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

Percentage Accurate: 45.3% → 81.6%

Time: 4.4s

Alternatives: 8

Speedup: 11.9×

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: 45.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 8 \cdot 10^{+107}:\\ \;\;\;\;\left(t\_0 + \left(r - p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\_m\right)\\ \end{array} \end{array} \]

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 8e+107) (* (+ 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 <= 8e+107) {
		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 <= 8e+107)
		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, 8e+107], 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 < 7.9999999999999998e107

   1. Initial program 57.8%

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

   2. Taylor expanded in p around -inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{-1 \cdot \left(p \cdot \left(1 + -1 \cdot \frac{r}{p}\right)\right)}\right) \]
   3. 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. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 73.5

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

   4. Applied rewrites 73.5%

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

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 84.6

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

   9. Applied rewrites 84.6%

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

   if 7.9999999999999998e107 < q

   1. Initial program 19.9%

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

   2. Taylor expanded in p around -inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{-1 \cdot \left(p \cdot \left(1 + -1 \cdot \frac{r}{p}\right)\right)}\right) \]
   3. 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. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 27.1

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

   4. Applied rewrites 27.1%

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

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 32.2

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

   7. Applied rewrites 32.2%

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

   8. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   9. 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

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

   10. Applied rewrites 75.4%

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

   11. Taylor expanded in q around 0

       \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   12. 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-+.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-fabs.f64 75.4

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

   13. Applied rewrites 75.4%

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

Alternative 2: 64.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;p \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;\left(t\_0 + \left(-p\right)\right) \cdot 0.5\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= p -1.45e-12)
     (* (+ t_0 (- p)) 0.5)
     (if (<= p 2e-109) (fma t_0 0.5 q_m) r))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (p <= -1.45e-12) {
		tmp = (t_0 + -p) * 0.5;
	} else if (p <= 2e-109) {
		tmp = fma(t_0, 0.5, q_m);
	} else {
		tmp = r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (p <= -1.45e-12)
		tmp = Float64(Float64(t_0 + Float64(-p)) * 0.5);
	elseif (p <= 2e-109)
		tmp = fma(t_0, 0.5, q_m);
	else
		tmp = r;
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, -1.45e-12], N[(N[(t$95$0 + (-p)), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[p, 2e-109], N[(t$95$0 * 0.5 + q$95$m), $MachinePrecision], r]]]

Derivation

1. Split input into 3 regimes

2. if p < -1.4500000000000001e-12

   1. Initial program 34.9%

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

   2. Taylor expanded in p around -inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{-1 \cdot \left(p \cdot \left(1 + -1 \cdot \frac{r}{p}\right)\right)}\right) \]
   3. 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. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 78.1

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

   4. Applied rewrites 78.1%

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

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 78.1

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

   7. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 78.1

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

   9. Applied rewrites 78.1%

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

   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. lower-neg.f64 67.4

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

   12. Applied rewrites 67.4%

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

   if -1.4500000000000001e-12 < p < 2e-109

   1. Initial program 59.1%

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

   2. Taylor expanded in p around -inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{-1 \cdot \left(p \cdot \left(1 + -1 \cdot \frac{r}{p}\right)\right)}\right) \]
   3. 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. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 33.4

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

   4. Applied rewrites 33.4%

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

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 52.9

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   9. 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

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

   10. Applied rewrites 56.8%

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

   11. Taylor expanded in q around 0

       \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   12. 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-+.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-fabs.f64 58.2

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

   13. Applied rewrites 58.2%

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

   if 2e-109 < p

   1. Initial program 34.9%

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

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 16.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 16.0%

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

   7. Taylor expanded in p around -inf

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

   9. Applied rewrites 78.2%

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

Alternative 3: 59.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;r \leq 7.3 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + r\right) \cdot 0.5\\ \end{array} \end{array} \]

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 (<= r 7.3e+130) (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 (r <= 7.3e+130) {
		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 (r <= 7.3e+130)
		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[r, 7.3e+130], 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 2 regimes

2. if r < 7.3e130

   1. Initial program 56.3%

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

   2. Taylor expanded in p around -inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{-1 \cdot \left(p \cdot \left(1 + -1 \cdot \frac{r}{p}\right)\right)}\right) \]
   3. 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. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 58.6

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

   4. Applied rewrites 58.6%

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

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   9. 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

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

   10. Applied rewrites 50.1%

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

   11. Taylor expanded in q around 0

       \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   12. 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-+.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-fabs.f64 51.9

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

   13. Applied rewrites 51.9%

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

   if 7.3e130 < r

   1. Initial program 14.3%

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

   2. Taylor expanded in p around -inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{-1 \cdot \left(p \cdot \left(1 + -1 \cdot \frac{r}{p}\right)\right)}\right) \]
   3. 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. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 57.2

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

   4. Applied rewrites 57.2%

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

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 86.6

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

   7. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 86.6

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

   9. Applied rewrites 86.6%

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

   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 79.9%

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

Alternative 4: 59.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 7.3 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 7.3e+130) (fma (+ (fabs r) (fabs p)) 0.5 q_m) r))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 7.3e+130) {
		tmp = fma((fabs(r) + fabs(p)), 0.5, q_m);
	} else {
		tmp = r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 7.3e+130)
		tmp = fma(Float64(abs(r) + abs(p)), 0.5, q_m);
	else
		tmp = r;
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 7.3e+130], N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision], r]

Derivation

1. Split input into 2 regimes

2. if r < 7.3e130

   1. Initial program 56.3%

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

   2. Taylor expanded in p around -inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{-1 \cdot \left(p \cdot \left(1 + -1 \cdot \frac{r}{p}\right)\right)}\right) \]
   3. 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. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 58.6

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

   4. Applied rewrites 58.6%

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

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   9. 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

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

   10. Applied rewrites 50.1%

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

   11. Taylor expanded in q around 0

       \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   12. 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-+.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-fabs.f64 51.9

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

   13. Applied rewrites 51.9%

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

   if 7.3e130 < r

   1. Initial program 14.3%

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

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 16.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 15.8%

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

   7. Taylor expanded in p around -inf

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

   9. Applied rewrites 79.5%

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

Alternative 5: 53.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 7.3 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(r + p, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 7.3e+130) (fma (+ r p) 0.5 q_m) r))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 7.3e+130) {
		tmp = fma((r + p), 0.5, q_m);
	} else {
		tmp = r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 7.3e+130)
		tmp = fma(Float64(r + p), 0.5, q_m);
	else
		tmp = r;
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 7.3e+130], N[(N[(r + p), $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision], r]

Derivation

1. Split input into 2 regimes

2. if r < 7.3e130

   1. Initial program 56.3%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval 43.9

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

   7. Applied rewrites 43.9%

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

   if 7.3e130 < r

   1. Initial program 14.3%

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

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 16.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 15.8%

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

   7. Taylor expanded in p around -inf

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

   9. Applied rewrites 79.5%

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

Alternative 6: 52.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 7.2 \cdot 10^{+130}:\\ \;\;\;\;q\_m\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (if (<= r 7.2e+130) q_m r))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 7.2e+130) {
		tmp = q_m;
	} else {
		tmp = r;
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (r <= 7.2d+130) then
        tmp = q_m
    else
        tmp = r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 7.2e+130) {
		tmp = q_m;
	} else {
		tmp = r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if r <= 7.2e+130:
		tmp = q_m
	else:
		tmp = r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 7.2e+130)
		tmp = q_m;
	else
		tmp = r;
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (r <= 7.2e+130)
		tmp = q_m;
	else
		tmp = r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 7.2e+130], q$95$m, r]

Derivation

1. Split input into 2 regimes

2. if r < 7.2000000000000002e130

   1. Initial program 56.3%

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

   2. Taylor expanded in q around inf

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

   4. Applied rewrites 43.2%

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

   if 7.2000000000000002e130 < r

   1. Initial program 14.3%

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

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 16.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 15.8%

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

   7. Taylor expanded in p around -inf

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

   9. Applied rewrites 79.5%

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

Alternative 7: 34.6% accurate, 56.9× speedup

Math

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

FPCore

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

C

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

Fortran

q_m =     private
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

2. Taylor expanded in p around inf

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

4. Applied rewrites 8.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in p around -inf

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

9. Applied rewrites 34.6%

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

Alternative 8: 1.9% accurate, 56.9× speedup

Math

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

FPCore

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

C

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

Fortran

q_m =     private
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

2. Taylor expanded in p around inf

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

4. Applied rewrites 8.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in p around inf

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

9. Applied rewrites 1.9%

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

Reproduce

herbie shell --seed 2025124 
(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)))))))