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

Percentage Accurate: 45.2% → 80.0%

Time: 5.0s

Alternatives: 7

Speedup: 12.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 45.2% 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: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) + \left(\left(-p\right) + r\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0
         (*
          (/ 1.0 2.0)
          (+
           (+ (fabs p) (fabs r))
           (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0))))))))
   (if (<= t_0 2e+148) t_0 (* (+ (+ (fabs r) (fabs p)) (+ (- p) r)) 0.5))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = (1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
	double tmp;
	if (t_0 <= 2e+148) {
		tmp = t_0;
	} else {
		tmp = ((fabs(r) + fabs(p)) + (-p + r)) * 0.5;
	}
	return tmp;
}

Fortran

NOTE: p, r, and q 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)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) + sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q ** 2.0d0)))))
    if (t_0 <= 2d+148) then
        tmp = t_0
    else
        tmp = ((abs(r) + abs(p)) + (-p + r)) * 0.5d0
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double t_0 = (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)))));
	double tmp;
	if (t_0 <= 2e+148) {
		tmp = t_0;
	} else {
		tmp = ((Math.abs(r) + Math.abs(p)) + (-p + r)) * 0.5;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	t_0 = (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)))))
	tmp = 0
	if t_0 <= 2e+148:
		tmp = t_0
	else:
		tmp = ((math.fabs(r) + math.fabs(p)) + (-p + r)) * 0.5
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = 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))))))
	tmp = 0.0
	if (t_0 <= 2e+148)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) + Float64(Float64(-p) + r)) * 0.5);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	t_0 = (1.0 / 2.0) * ((abs(p) + abs(r)) + sqrt((((p - r) ^ 2.0) + (4.0 * (q ^ 2.0)))));
	tmp = 0.0;
	if (t_0 <= 2e+148)
		tmp = t_0;
	else
		tmp = ((abs(r) + abs(p)) + (-p + r)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 2e+148], t$95$0, N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[((-p) + r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))))))) < 2.0000000000000001e148

   1. Initial program 96.7%

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

   if 2.0000000000000001e148 < (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))))))

   1. Initial program 8.2%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 22.7

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

   5. Applied rewrites 22.7%

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

   6. Taylor expanded in p around 0

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r + \color{blue}{-1 \cdot p}\right)\right) \]
   7. 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. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 30.2

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

   8. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 30.2

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

   10. Applied rewrites 30.2%

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

       1. lift-fma.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-+.f64 N/A

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

       4. lift-neg.f64 30.2

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

   12. Applied rewrites 30.2%

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

Alternative 2: 56.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;p \leq -2.35 \cdot 10^{+42}:\\ \;\;\;\;\left(\left(-p\right) + t\_0\right) \cdot 0.5\\ \mathbf{elif}\;p \leq 5.8 \cdot 10^{-208}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(q, \frac{q}{r}, r\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= p -2.35e+42)
     (* (+ (- p) t_0) 0.5)
     (if (<= p 5.8e-208) (fma t_0 0.5 q) (fma q (/ q r) r)))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (p <= -2.35e+42) {
		tmp = (-p + t_0) * 0.5;
	} else if (p <= 5.8e-208) {
		tmp = fma(t_0, 0.5, q);
	} else {
		tmp = fma(q, (q / r), r);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (p <= -2.35e+42)
		tmp = Float64(Float64(Float64(-p) + t_0) * 0.5);
	elseif (p <= 5.8e-208)
		tmp = fma(t_0, 0.5, q);
	else
		tmp = fma(q, Float64(q / r), r);
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, -2.35e+42], N[(N[((-p) + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[p, 5.8e-208], N[(t$95$0 * 0.5 + q), $MachinePrecision], N[(q * N[(q / r), $MachinePrecision] + r), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < -2.34999999999999993e42

   1. Initial program 29.9%

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

   3. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.9

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

   5. Applied rewrites 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 72.9%

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

   if -2.34999999999999993e42 < p < 5.7999999999999999e-208

   1. Initial program 57.9%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 19.0

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   8. Applied rewrites 34.4%

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

   9. Taylor expanded in q around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lift-fabs.f64 N/A

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

       6. lift-fabs.f64 N/A

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

       7. lift-+.f64 36.2

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

   11. Applied rewrites 36.2%

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

   if 5.7999999999999999e-208 < p

   1. Initial program 39.0%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 18.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(r, 2, p\right), 0.5, \frac{q \cdot q}{r}\right) \]

   8. Applied rewrites 18.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(r, 2, p\right), \color{blue}{0.5}, \frac{q \cdot q}{r}\right) \]

   9. Taylor expanded in p around 0

      \[\leadsto r + \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{{q}^{2}}{r} + r \]

       2. pow2 N/A

          \[\leadsto \frac{q \cdot q}{r} + r \]

       3. associate-/l* N/A

          \[\leadsto q \cdot \frac{q}{r} + r \]

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 14.2

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

   11. Applied rewrites 14.2%

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

Alternative 3: 74.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q \leq 7.6 \cdot 10^{+95}:\\ \;\;\;\;\left(t\_0 + \left(\left(-p\right) + r\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= q 7.6e+95) (* (+ t_0 (+ (- p) r)) 0.5) (fma t_0 0.5 q))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (q <= 7.6e+95) {
		tmp = (t_0 + (-p + r)) * 0.5;
	} else {
		tmp = fma(t_0, 0.5, q);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (q <= 7.6e+95)
		tmp = Float64(Float64(t_0 + Float64(Float64(-p) + r)) * 0.5);
	else
		tmp = fma(t_0, 0.5, q);
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q, 7.6e+95], N[(N[(t$95$0 + N[((-p) + r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(t$95$0 * 0.5 + q), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 7.5999999999999999e95

   1. Initial program 49.1%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 28.6

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in p around 0

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r + \color{blue}{-1 \cdot p}\right)\right) \]
   7. 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. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 34.2

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

   8. Applied rewrites 34.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 34.2

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

   10. Applied rewrites 34.2%

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

       1. lift-fma.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-+.f64 N/A

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

       4. lift-neg.f64 34.2

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

   12. Applied rewrites 34.2%

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

   if 7.5999999999999999e95 < q

   1. Initial program 22.9%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in q around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lift-fabs.f64 N/A

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

       6. lift-fabs.f64 N/A

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

       7. lift-+.f64 69.5

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

   11. Applied rewrites 69.5%

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

Alternative 4: 46.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 2.6 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, q\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(q, \frac{q}{r}, r\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r 2.6e+93) (fma (+ (fabs r) (fabs p)) 0.5 q) (fma q (/ q r) r)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 2.6e+93) {
		tmp = fma((fabs(r) + fabs(p)), 0.5, q);
	} else {
		tmp = fma(q, (q / r), r);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 2.6e+93)
		tmp = fma(Float64(abs(r) + abs(p)), 0.5, q);
	else
		tmp = fma(q, Float64(q / r), r);
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[r, 2.6e+93], N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + q), $MachinePrecision], N[(q * N[(q / r), $MachinePrecision] + r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.6e93

   1. Initial program 48.9%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 25.9

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

   5. Applied rewrites 25.9%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   8. Applied rewrites 28.7%

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

   9. Taylor expanded in q around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lift-fabs.f64 N/A

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

       6. lift-fabs.f64 N/A

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

       7. lift-+.f64 30.9

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

   11. Applied rewrites 30.9%

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

   if 2.6e93 < r

   1. Initial program 15.6%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 52.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(r, 2, p\right), 0.5, \frac{q \cdot q}{r}\right) \]

   8. Applied rewrites 52.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(r, 2, p\right), \color{blue}{0.5}, \frac{q \cdot q}{r}\right) \]

   9. Taylor expanded in p around 0

      \[\leadsto r + \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{{q}^{2}}{r} + r \]

       2. pow2 N/A

          \[\leadsto \frac{q \cdot q}{r} + r \]

       3. associate-/l* N/A

          \[\leadsto q \cdot \frac{q}{r} + r \]

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 60.1

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

   11. Applied rewrites 60.1%

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

Alternative 5: 45.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 2.6 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, q\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r 2.6e+93) (fma (+ (fabs r) (fabs p)) 0.5 q) r))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 2.6e+93) {
		tmp = fma((fabs(r) + fabs(p)), 0.5, q);
	} else {
		tmp = r;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 2.6e+93)
		tmp = fma(Float64(abs(r) + abs(p)), 0.5, q);
	else
		tmp = r;
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[r, 2.6e+93], N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + q), $MachinePrecision], r]

Derivation

1. Split input into 2 regimes

2. if r < 2.6e93

   1. Initial program 48.9%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 25.9

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

   5. Applied rewrites 25.9%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   8. Applied rewrites 28.7%

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

   9. Taylor expanded in q around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lift-fabs.f64 N/A

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

       6. lift-fabs.f64 N/A

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

       7. lift-+.f64 30.9

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

   11. Applied rewrites 30.9%

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

   if 2.6e93 < r

   1. Initial program 15.6%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 59.6%

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

Alternative 6: 41.3% accurate, 35.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;q\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (if (<= r 8.5e+47) q r))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 8.5e+47) {
		tmp = q;
	} else {
		tmp = r;
	}
	return tmp;
}

Fortran

NOTE: p, r, and q 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)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (r <= 8.5d+47) then
        tmp = q
    else
        tmp = r
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (r <= 8.5e+47) {
		tmp = q;
	} else {
		tmp = r;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 8.5e+47:
		tmp = q
	else:
		tmp = r
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 8.5e+47)
		tmp = q;
	else
		tmp = r;
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 8.5e+47)
		tmp = q;
	else
		tmp = r;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[r, 8.5e+47], q, r]

Derivation

1. Split input into 2 regimes

2. if r < 8.5000000000000008e47

   1. Initial program 48.0%

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

   3. Taylor expanded in q around inf

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

   5. Applied rewrites 20.0%

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

   if 8.5000000000000008e47 < r

   1. Initial program 26.1%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 24.6%

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

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 55.4%

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

Alternative 7: 18.3% accurate, 250.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: p, r, and q 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)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = q
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	return q;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	return q

Julia

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

MATLAB

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

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := q

Derivation

1. Initial program 44.2%

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

3. Taylor expanded in q around inf

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

5. Applied rewrites 19.4%

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

Reproduce

herbie shell --seed 2025073 
(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)))))))