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

Percentage Accurate: 45.4% → 82.7%

Time: 6.8s

Alternatives: 11

Speedup: 16.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 45.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;4 \cdot {q\_m}^{2} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (* 4.0 (pow q_m 2.0)) 2e+216)
   (* 0.5 (+ (+ (fabs p) (fabs r)) (- r p)))
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if ((4.0 * pow(q_m, 2.0)) <= 2e+216) {
		tmp = 0.5 * ((fabs(p) + fabs(r)) + (r - p));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (Float64(4.0 * (q_m ^ 2.0)) <= 2e+216)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + abs(r)) + Float64(r - p)));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2e+216], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(r - p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 2e216

   1. Initial program 55.4%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 38.0

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

   5. Applied rewrites 38.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 49.8

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

   10. Applied rewrites 49.8%

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

   if 2e216 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 20.1%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 40.4%

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

Alternative 2: 82.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;4 \cdot {q\_m}^{2} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;-0.5 \cdot \left(p - \left(\left(r + \left|r\right|\right) + \left|p\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (* 4.0 (pow q_m 2.0)) 2e+216)
   (* -0.5 (- p (+ (+ r (fabs r)) (fabs p))))
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if ((4.0 * pow(q_m, 2.0)) <= 2e+216) {
		tmp = -0.5 * (p - ((r + fabs(r)) + fabs(p)));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (Float64(4.0 * (q_m ^ 2.0)) <= 2e+216)
		tmp = Float64(-0.5 * Float64(p - Float64(Float64(r + abs(r)) + abs(p))));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2e+216], N[(-0.5 * N[(p - N[(N[(r + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 2e216

   1. Initial program 55.4%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 50.0%

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

   if 2e216 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 20.1%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 40.4%

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

Alternative 3: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;4 \cdot {q\_m}^{2} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + r\right) + \left(r - p\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (* 4.0 (pow q_m 2.0)) 2e+216)
   (* 0.5 (+ (+ (fabs p) r) (- r p)))
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if ((4.0 * pow(q_m, 2.0)) <= 2e+216) {
		tmp = 0.5 * ((fabs(p) + r) + (r - p));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (Float64(4.0 * (q_m ^ 2.0)) <= 2e+216)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + r) + Float64(r - p)));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2e+216], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + r), $MachinePrecision] + N[(r - p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 2e216

   1. Initial program 55.4%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 38.0

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

   5. Applied rewrites 38.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 49.8

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

   10. Applied rewrites 49.8%

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

       1. lift-fabs.f64 N/A

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

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

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

       3. sqrt-unprod N/A

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

       4. rem-square-sqrt 48.4

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

   12. Applied rewrites 48.4%

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

   if 2e216 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 20.1%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 40.4%

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

Alternative 4: 64.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 6.6 \cdot 10^{-195}:\\ \;\;\;\;-0.5 \cdot \left(\left(p - \left|p\right|\right) - \left|r\right|\right)\\ \mathbf{elif}\;r \leq 7 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(r + \left|p\right|\right) + \left|r\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 6.6e-195)
   (* -0.5 (- (- p (fabs p)) (fabs r)))
   (if (<= r 7e+58)
     (fma 0.5 (+ (fabs r) (fabs p)) q_m)
     (* (+ (+ r (fabs p)) (fabs r)) 0.5))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 6.6e-195) {
		tmp = -0.5 * ((p - fabs(p)) - fabs(r));
	} else if (r <= 7e+58) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	} else {
		tmp = ((r + fabs(p)) + fabs(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)
	tmp = 0.0
	if (r <= 6.6e-195)
		tmp = Float64(-0.5 * Float64(Float64(p - abs(p)) - abs(r)));
	elseif (r <= 7e+58)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	else
		tmp = Float64(Float64(Float64(r + abs(p)) + abs(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_] := If[LessEqual[r, 6.6e-195], N[(-0.5 * N[(N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 7e+58], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], N[(N[(N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 6.6e-195

   1. Initial program 47.6%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 27.2

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

   5. Applied rewrites 27.2%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 31.3%

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

   if 6.6e-195 < r < 6.9999999999999995e58

   1. Initial program 43.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 28.1%

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

   if 6.9999999999999995e58 < r

   1. Initial program 29.6%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 75.7%

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

Alternative 5: 64.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 6.6 \cdot 10^{-195}:\\ \;\;\;\;-0.5 \cdot \left(\left(p - \left|p\right|\right) - \left|r\right|\right)\\ \mathbf{elif}\;r \leq 1.4 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

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 6.6e-195)
   (* -0.5 (- (- p (fabs p)) (fabs r)))
   (if (<= r 1.4e+59) (fma 0.5 (+ (fabs r) (fabs p)) 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 <= 6.6e-195) {
		tmp = -0.5 * ((p - fabs(p)) - fabs(r));
	} else if (r <= 1.4e+59) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), 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 <= 6.6e-195)
		tmp = Float64(-0.5 * Float64(Float64(p - abs(p)) - abs(r)));
	elseif (r <= 1.4e+59)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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, 6.6e-195], N[(-0.5 * N[(N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.4e+59], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], r]]

Derivation

1. Split input into 3 regimes

2. if r < 6.6e-195

   1. Initial program 47.6%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 27.2

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

   5. Applied rewrites 27.2%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 31.3%

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

   if 6.6e-195 < r < 1.3999999999999999e59

   1. Initial program 43.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 28.1%

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

   if 1.3999999999999999e59 < r

   1. Initial program 29.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-prod N/A

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

      6. lower-fma.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. sqrt-prod N/A

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

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

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

      10. lift-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-fabs.f64 N/A

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

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

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

      14. sqrt-prod N/A

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

      15. rem-square-sqrt N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-prod N/A

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

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

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

      19. lift-fabs.f64 N/A

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

      20. lower-sqrt.f64 81.5

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

      21. lift-fabs.f64 N/A

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

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

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt 81.5

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

      25. lift-fabs.f64 N/A

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

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

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

      27. pow2 N/A

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

      28. sqrt-pow1 N/A

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

      29. metadata-eval N/A

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

      30. unpow1 74.4

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

   10. Applied rewrites 74.4%

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

   11. Taylor expanded in r around inf

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

       1. div-add N/A

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

       2. associate-*r/ N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \left(\color{blue}{0} \cdot \frac{p}{r}\right)\right) \]

       6. mul0-lft N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{0}\right) \]

       7. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \color{blue}{0}\right) \]

       8. metadata-eval N/A

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

       9. *-rgt-identity 75.2

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

   13. Applied rewrites 75.2%

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

Alternative 6: 64.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 6.6 \cdot 10^{-195}:\\ \;\;\;\;-0.5 \cdot \left(\left(p - \left|p\right|\right) - r\right)\\ \mathbf{elif}\;r \leq 1.4 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

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 6.6e-195)
   (* -0.5 (- (- p (fabs p)) r))
   (if (<= r 1.4e+59) (fma 0.5 (+ (fabs r) (fabs p)) 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 <= 6.6e-195) {
		tmp = -0.5 * ((p - fabs(p)) - r);
	} else if (r <= 1.4e+59) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), 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 <= 6.6e-195)
		tmp = Float64(-0.5 * Float64(Float64(p - abs(p)) - r));
	elseif (r <= 1.4e+59)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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, 6.6e-195], N[(-0.5 * N[(N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.4e+59], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], r]]

Derivation

1. Split input into 3 regimes

2. if r < 6.6e-195

   1. Initial program 47.6%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 27.2

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

   5. Applied rewrites 27.2%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 31.3%

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

   13. Applied rewrites 25.2%

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

   if 6.6e-195 < r < 1.3999999999999999e59

   1. Initial program 43.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 28.1%

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

   if 1.3999999999999999e59 < r

   1. Initial program 29.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-prod N/A

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

      6. lower-fma.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. sqrt-prod N/A

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

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

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

      10. lift-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-fabs.f64 N/A

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

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

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

      14. sqrt-prod N/A

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

      15. rem-square-sqrt N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-prod N/A

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

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

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

      19. lift-fabs.f64 N/A

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

      20. lower-sqrt.f64 81.5

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

      21. lift-fabs.f64 N/A

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

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

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt 81.5

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

      25. lift-fabs.f64 N/A

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

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

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

      27. pow2 N/A

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

      28. sqrt-pow1 N/A

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

      29. metadata-eval N/A

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

      30. unpow1 74.4

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

   10. Applied rewrites 74.4%

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

   11. Taylor expanded in r around inf

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

       1. div-add N/A

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

       2. associate-*r/ N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \left(\color{blue}{0} \cdot \frac{p}{r}\right)\right) \]

       6. mul0-lft N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{0}\right) \]

       7. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \color{blue}{0}\right) \]

       8. metadata-eval N/A

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

       9. *-rgt-identity 75.2

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

   13. Applied rewrites 75.2%

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

Alternative 7: 62.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 4.2 \cdot 10^{-190}:\\ \;\;\;\;-0.5 \cdot \left(\left(p - \left|p\right|\right) - r\right)\\ \mathbf{elif}\;r \leq 7 \cdot 10^{+58}:\\ \;\;\;\;\left(q\_m + q\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 4.2e-190)
   (* -0.5 (- (- p (fabs p)) r))
   (if (<= r 7e+58) (* (+ q_m q_m) 0.5) r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 4.2e-190) {
		tmp = -0.5 * ((p - fabs(p)) - r);
	} else if (r <= 7e+58) {
		tmp = (q_m + q_m) * 0.5;
	} else {
		tmp = r;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (r <= 4.2d-190) then
        tmp = (-0.5d0) * ((p - abs(p)) - r)
    else if (r <= 7d+58) then
        tmp = (q_m + q_m) * 0.5d0
    else
        tmp = r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 4.2e-190) {
		tmp = -0.5 * ((p - Math.abs(p)) - r);
	} else if (r <= 7e+58) {
		tmp = (q_m + q_m) * 0.5;
	} else {
		tmp = r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if r <= 4.2e-190:
		tmp = -0.5 * ((p - math.fabs(p)) - r)
	elif r <= 7e+58:
		tmp = (q_m + q_m) * 0.5
	else:
		tmp = r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 4.2e-190)
		tmp = Float64(-0.5 * Float64(Float64(p - abs(p)) - r));
	elseif (r <= 7e+58)
		tmp = Float64(Float64(q_m + q_m) * 0.5);
	else
		tmp = r;
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (r <= 4.2e-190)
		tmp = -0.5 * ((p - abs(p)) - r);
	elseif (r <= 7e+58)
		tmp = (q_m + q_m) * 0.5;
	else
		tmp = r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 4.2e-190], N[(-0.5 * N[(N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 7e+58], N[(N[(q$95$m + q$95$m), $MachinePrecision] * 0.5), $MachinePrecision], r]]

Derivation

1. Split input into 3 regimes

2. if r < 4.19999999999999983e-190

   1. Initial program 47.4%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 27.7

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

   5. Applied rewrites 27.7%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 27.7%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 31.7%

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

   13. Applied rewrites 25.7%

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

   if 4.19999999999999983e-190 < r < 6.9999999999999995e58

   1. Initial program 44.0%

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

   3. Taylor expanded in q around inf

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

      1. lower-*.f64 20.2

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

   5. Applied rewrites 20.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.2

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

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(q \cdot 2\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(q \cdot 2\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   7. Applied rewrites 20.2%

      \[\leadsto \color{blue}{\left(q \cdot 2\right) \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 20.2%

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

   if 6.9999999999999995e58 < r

   1. Initial program 29.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-prod N/A

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

      6. lower-fma.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. sqrt-prod N/A

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

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

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

      10. lift-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-fabs.f64 N/A

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

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

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

      14. sqrt-prod N/A

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

      15. rem-square-sqrt N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-prod N/A

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

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

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

      19. lift-fabs.f64 N/A

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

      20. lower-sqrt.f64 81.5

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

      21. lift-fabs.f64 N/A

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

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

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt 81.5

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

      25. lift-fabs.f64 N/A

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

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

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

      27. pow2 N/A

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

      28. sqrt-pow1 N/A

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

      29. metadata-eval N/A

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

      30. unpow1 74.4

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

   10. Applied rewrites 74.4%

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

   11. Taylor expanded in r around inf

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

       1. div-add N/A

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

       2. associate-*r/ N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \left(\color{blue}{0} \cdot \frac{p}{r}\right)\right) \]

       6. mul0-lft N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{0}\right) \]

       7. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \color{blue}{0}\right) \]

       8. metadata-eval N/A

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

       9. *-rgt-identity 75.2

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

   13. Applied rewrites 75.2%

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

Alternative 8: 55.0% accurate, 16.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{+58}:\\ \;\;\;\;\left(q\_m + q\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

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 7e+58) (* (+ q_m q_m) 0.5) r))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 7e+58) {
		tmp = (q_m + q_m) * 0.5;
	} 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 <= 7d+58) then
        tmp = (q_m + q_m) * 0.5d0
    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 <= 7e+58) {
		tmp = (q_m + q_m) * 0.5;
	} 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 <= 7e+58:
		tmp = (q_m + q_m) * 0.5
	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 <= 7e+58)
		tmp = Float64(Float64(q_m + q_m) * 0.5);
	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 <= 7e+58)
		tmp = (q_m + q_m) * 0.5;
	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, 7e+58], N[(N[(q$95$m + q$95$m), $MachinePrecision] * 0.5), $MachinePrecision], r]

Derivation

1. Split input into 2 regimes

2. if r < 6.9999999999999995e58

   1. Initial program 46.7%

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

   3. Taylor expanded in q around inf

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

      1. lower-*.f64 18.2

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

   5. Applied rewrites 18.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 18.2

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

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(q \cdot 2\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(q \cdot 2\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   7. Applied rewrites 18.2%

      \[\leadsto \color{blue}{\left(q \cdot 2\right) \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.2%

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

   if 6.9999999999999995e58 < r

   1. Initial program 29.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-prod N/A

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

      6. lower-fma.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. sqrt-prod N/A

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

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

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

      10. lift-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-fabs.f64 N/A

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

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

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

      14. sqrt-prod N/A

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

      15. rem-square-sqrt N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-prod N/A

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

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

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

      19. lift-fabs.f64 N/A

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

      20. lower-sqrt.f64 81.5

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

      21. lift-fabs.f64 N/A

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

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

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt 81.5

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

      25. lift-fabs.f64 N/A

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

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

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

      27. pow2 N/A

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

      28. sqrt-pow1 N/A

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

      29. metadata-eval N/A

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

      30. unpow1 74.4

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

   10. Applied rewrites 74.4%

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

   11. Taylor expanded in r around inf

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

       1. div-add N/A

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

       2. associate-*r/ N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \left(\color{blue}{0} \cdot \frac{p}{r}\right)\right) \]

       6. mul0-lft N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{0}\right) \]

       7. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \color{blue}{0}\right) \]

       8. metadata-eval N/A

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

       9. *-rgt-identity 75.2

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

   13. Applied rewrites 75.2%

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

Alternative 9: 39.1% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 3.6 \cdot 10^{-154}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

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 3.6e-154) (* -0.5 p) 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 <= 3.6e-154) {
		tmp = -0.5 * p;
	} 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 <= 3.6d-154) then
        tmp = (-0.5d0) * p
    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 <= 3.6e-154) {
		tmp = -0.5 * p;
	} 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 <= 3.6e-154:
		tmp = -0.5 * p
	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 <= 3.6e-154)
		tmp = Float64(-0.5 * p);
	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 <= 3.6e-154)
		tmp = -0.5 * p;
	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, 3.6e-154], N[(-0.5 * p), $MachinePrecision], r]

Derivation

1. Split input into 2 regimes

2. if r < 3.6000000000000003e-154

   1. Initial program 46.0%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   if 3.6000000000000003e-154 < r

   1. Initial program 37.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 62.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-prod N/A

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

      6. lower-fma.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. sqrt-prod N/A

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

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

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

      10. lift-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-fabs.f64 N/A

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

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

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

      14. sqrt-prod N/A

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

      15. rem-square-sqrt N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-prod N/A

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

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

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

      19. lift-fabs.f64 N/A

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

      20. lower-sqrt.f64 61.9

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

      21. lift-fabs.f64 N/A

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

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

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt 61.9

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

      25. lift-fabs.f64 N/A

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

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

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

      27. pow2 N/A

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

      28. sqrt-pow1 N/A

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

      29. metadata-eval N/A

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

      30. unpow1 52.0

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

   10. Applied rewrites 52.0%

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

   11. Taylor expanded in r around inf

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

       1. div-add N/A

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

       2. associate-*r/ N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \left(\color{blue}{0} \cdot \frac{p}{r}\right)\right) \]

       6. mul0-lft N/A

          \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{0}\right) \]

       7. metadata-eval N/A

          \[\leadsto r \cdot \left(1 + \color{blue}{0}\right) \]

       8. metadata-eval N/A

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

       9. *-rgt-identity 53.0

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

   13. Applied rewrites 53.0%

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

Alternative 10: 35.1% accurate, 250.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.0%

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

3. Taylor expanded in r around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 31.2

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

5. Applied rewrites 31.2%

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

6. Taylor expanded in p around 0

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

8. Applied rewrites 39.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fabs.f64 N/A

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

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

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

   5. sqrt-prod N/A

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

   6. lower-fma.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. sqrt-prod N/A

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

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

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

   10. lift-fabs.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-fabs.f64 N/A

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

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

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

   14. sqrt-prod N/A

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

   15. rem-square-sqrt N/A

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

   16. rem-square-sqrt N/A

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

   17. sqrt-prod N/A

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

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

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

   19. lift-fabs.f64 N/A

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

   20. lower-sqrt.f64 25.9

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

   21. lift-fabs.f64 N/A

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

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

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

   23. sqrt-prod N/A

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

   24. rem-square-sqrt 25.9

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

   25. lift-fabs.f64 N/A

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

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

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

   27. pow2 N/A

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

   28. sqrt-pow1 N/A

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

   29. metadata-eval N/A

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

   30. unpow1 18.5

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

10. Applied rewrites 18.5%

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

11. Taylor expanded in r around inf

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

    1. div-add N/A

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

    2. associate-*r/ N/A

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

    3. +-commutative N/A

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

    4. distribute-lft1-in N/A

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

    5. metadata-eval N/A

       \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \left(\color{blue}{0} \cdot \frac{p}{r}\right)\right) \]

    6. mul0-lft N/A

       \[\leadsto r \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{0}\right) \]

    7. metadata-eval N/A

       \[\leadsto r \cdot \left(1 + \color{blue}{0}\right) \]

    8. metadata-eval N/A

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

    9. *-rgt-identity 19.6

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

13. Applied rewrites 19.6%

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

Alternative 11: 2.3% accurate, 250.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ 0 \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 0.0)

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return 0.0;
}

Fortran

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

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

real(8) function code(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = 0.0d0
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	return 0.0;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return 0.0

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return 0.0
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = 0.0;
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_] := 0.0

Derivation

1. Initial program 43.0%

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

3. Taylor expanded in r around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 31.2

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

5. Applied rewrites 31.2%

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

6. Taylor expanded in p around 0

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

8. Applied rewrites 39.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fabs.f64 N/A

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

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

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

   5. sqrt-prod N/A

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

   6. lower-fma.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. sqrt-prod N/A

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

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

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

   10. lift-fabs.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-fabs.f64 N/A

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

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

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

   14. sqrt-prod N/A

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

   15. rem-square-sqrt N/A

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

   16. rem-square-sqrt N/A

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

   17. sqrt-prod N/A

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

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

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

   19. lift-fabs.f64 N/A

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

   20. lower-sqrt.f64 25.9

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

   21. lift-fabs.f64 N/A

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

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

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

   23. sqrt-prod N/A

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

   24. rem-square-sqrt 25.9

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

   25. lift-fabs.f64 N/A

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

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

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

   27. pow2 N/A

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

   28. sqrt-pow1 N/A

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

   29. metadata-eval N/A

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

   30. unpow1 18.5

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

10. Applied rewrites 18.5%

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

11. Taylor expanded in r around -inf

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

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. rem-square-sqrt N/A

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

    4. metadata-eval N/A

       \[\leadsto \frac{-1}{2} \cdot \left(\color{blue}{0} \cdot r\right) \]

    5. mul0-lft N/A

       \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

    6. metadata-eval 2.3

       \[\leadsto \color{blue}{0} \]

13. Applied rewrites 2.3%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(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)))))))