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

Percentage Accurate: 24.4% → 62.0%

Time: 4.6s

Alternatives: 8

Speedup: 83.3×

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: 24.4% 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: 62.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := 4 \cdot {q\_m}^{2}\\ \mathbf{if}\;t\_0 \leq 10^{-293}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left(\left|r\right| - r\right) + \left|p\right|\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{-4 \cdot \left(q\_m \cdot q\_m\right)}{\left(\left|r\right| + \left|p\right|\right) + \sqrt{\mathsf{fma}\left(q\_m \cdot q\_m, 4, r \cdot r\right)}} \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\left(-q\_m\right) \cdot \frac{q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow q_m 2.0))))
   (if (<= t_0 1e-293)
     (* 0.5 (+ p (+ (- (fabs r) r) (fabs p))))
     (if (<= t_0 2e+272)
       (*
        (/
         (* -4.0 (* q_m q_m))
         (+ (+ (fabs r) (fabs p)) (sqrt (fma (* q_m q_m) 4.0 (* r r)))))
        0.5)
       (if (<= t_0 2e+296) (* (- q_m) (/ q_m r)) (- q_m))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = 4.0 * pow(q_m, 2.0);
	double tmp;
	if (t_0 <= 1e-293) {
		tmp = 0.5 * (p + ((fabs(r) - r) + fabs(p)));
	} else if (t_0 <= 2e+272) {
		tmp = ((-4.0 * (q_m * q_m)) / ((fabs(r) + fabs(p)) + sqrt(fma((q_m * q_m), 4.0, (r * r))))) * 0.5;
	} else if (t_0 <= 2e+296) {
		tmp = -q_m * (q_m / r);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(4.0 * (q_m ^ 2.0))
	tmp = 0.0
	if (t_0 <= 1e-293)
		tmp = Float64(0.5 * Float64(p + Float64(Float64(abs(r) - r) + abs(p))));
	elseif (t_0 <= 2e+272)
		tmp = Float64(Float64(Float64(-4.0 * Float64(q_m * q_m)) / Float64(Float64(abs(r) + abs(p)) + sqrt(fma(Float64(q_m * q_m), 4.0, Float64(r * r))))) * 0.5);
	elseif (t_0 <= 2e+296)
		tmp = Float64(Float64(-q_m) * Float64(q_m / r));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-293], N[(0.5 * N[(p + N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+272], N[(N[(N[(-4.0 * N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(r * r), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+296], N[((-q$95$m) * N[(q$95$m / r), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1.0000000000000001e-293

   1. Initial program 25.7%

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

   3. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+r- N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-fabs.f64 72.0

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

   8. Applied rewrites 72.0%

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

   if 1.0000000000000001e-293 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 2.0000000000000001e272

   1. Initial program 31.4%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 29.3%

      \[\leadsto \color{blue}{\left(\left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 28.8%

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

   8. Taylor expanded in q around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 50.2

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

   10. Applied rewrites 50.2%

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

   if 2.0000000000000001e272 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1.99999999999999996e296

   1. Initial program 55.0%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in r around 0

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

      1. lower-*.f64 52.7

         \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - 2 \cdot q\right) \cdot 0.5 \]

   8. Applied rewrites 52.7%

      \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - 2 \cdot q\right) \cdot 0.5 \]

   9. Taylor expanded in q around 0

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

   11. Applied rewrites 22.0%

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

   12. Taylor expanded in r around 0

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

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot {q}^{2}}{r} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left({q}^{2}\right)}{r} \]

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(q \cdot q\right)}{r} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(q\right)\right) \cdot q}{r} \]

       5. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(q\right)\right) \cdot \frac{q}{r} \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(q\right)\right) \cdot \frac{q}{r} \]

       7. lift-neg.f64 N/A

          \[\leadsto \left(-q\right) \cdot \frac{q}{r} \]

       8. lower-/.f64 24.3

          \[\leadsto \left(-q\right) \cdot \frac{q}{r} \]

   14. Applied rewrites 24.3%

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

   if 1.99999999999999996e296 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 8.1%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(q\right) \]

      2. lower-neg.f64 75.5

         \[\leadsto -q \]

   5. Applied rewrites 75.5%

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

Alternative 2: 59.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := 4 \cdot {q\_m}^{2}\\ \mathbf{if}\;t\_0 \leq 10^{-148}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left(\left|r\right| - r\right) + \left|p\right|\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{-4 \cdot \left(q\_m \cdot q\_m\right)}{\mathsf{fma}\left(q\_m, 2, \left|r\right|\right) + \left|p\right|} \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\left(-q\_m\right) \cdot \frac{q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow q_m 2.0))))
   (if (<= t_0 1e-148)
     (* 0.5 (+ p (+ (- (fabs r) r) (fabs p))))
     (if (<= t_0 2e+272)
       (* (/ (* -4.0 (* q_m q_m)) (+ (fma q_m 2.0 (fabs r)) (fabs p))) 0.5)
       (if (<= t_0 2e+296) (* (- q_m) (/ q_m r)) (- q_m))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = 4.0 * pow(q_m, 2.0);
	double tmp;
	if (t_0 <= 1e-148) {
		tmp = 0.5 * (p + ((fabs(r) - r) + fabs(p)));
	} else if (t_0 <= 2e+272) {
		tmp = ((-4.0 * (q_m * q_m)) / (fma(q_m, 2.0, fabs(r)) + fabs(p))) * 0.5;
	} else if (t_0 <= 2e+296) {
		tmp = -q_m * (q_m / r);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(4.0 * (q_m ^ 2.0))
	tmp = 0.0
	if (t_0 <= 1e-148)
		tmp = Float64(0.5 * Float64(p + Float64(Float64(abs(r) - r) + abs(p))));
	elseif (t_0 <= 2e+272)
		tmp = Float64(Float64(Float64(-4.0 * Float64(q_m * q_m)) / Float64(fma(q_m, 2.0, abs(r)) + abs(p))) * 0.5);
	elseif (t_0 <= 2e+296)
		tmp = Float64(Float64(-q_m) * Float64(q_m / r));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-148], N[(0.5 * N[(p + N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+272], N[(N[(N[(-4.0 * N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(q$95$m * 2.0 + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+296], N[((-q$95$m) * N[(q$95$m / r), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 9.99999999999999936e-149

   1. Initial program 24.7%

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

   3. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+r- N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-fabs.f64 65.1

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

   8. Applied rewrites 65.1%

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

   if 9.99999999999999936e-149 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 2.0000000000000001e272

   1. Initial program 34.7%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 33.5%

      \[\leadsto \color{blue}{\left(\left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 32.9%

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

   8. Taylor expanded in q around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.6

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

   10. Applied rewrites 49.6%

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

   11. Taylor expanded in r around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-fabs.f64 N/A

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

       7. lift-fabs.f64 43.7

          \[\leadsto \frac{-4 \cdot \left(q \cdot q\right)}{\mathsf{fma}\left(q, 2, \left|r\right|\right) + \left|p\right|} \cdot 0.5 \]

   13. Applied rewrites 43.7%

       \[\leadsto \frac{-4 \cdot \left(q \cdot q\right)}{\mathsf{fma}\left(q, 2, \left|r\right|\right) + \left|p\right|} \cdot 0.5 \]

   if 2.0000000000000001e272 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1.99999999999999996e296

   1. Initial program 55.0%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in r around 0

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

      1. lower-*.f64 52.7

         \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - 2 \cdot q\right) \cdot 0.5 \]

   8. Applied rewrites 52.7%

      \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - 2 \cdot q\right) \cdot 0.5 \]

   9. Taylor expanded in q around 0

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

   11. Applied rewrites 22.0%

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

   12. Taylor expanded in r around 0

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

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot {q}^{2}}{r} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left({q}^{2}\right)}{r} \]

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(q \cdot q\right)}{r} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(q\right)\right) \cdot q}{r} \]

       5. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(q\right)\right) \cdot \frac{q}{r} \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(q\right)\right) \cdot \frac{q}{r} \]

       7. lift-neg.f64 N/A

          \[\leadsto \left(-q\right) \cdot \frac{q}{r} \]

       8. lower-/.f64 24.3

          \[\leadsto \left(-q\right) \cdot \frac{q}{r} \]

   14. Applied rewrites 24.3%

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

   if 1.99999999999999996e296 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 8.1%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(q\right) \]

      2. lower-neg.f64 75.5

         \[\leadsto -q \]

   5. Applied rewrites 75.5%

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

Alternative 3: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := 4 \cdot {q\_m}^{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-71}:\\ \;\;\;\;\left(-p\right) \cdot \left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p} \cdot -0.5 - 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{-4 \cdot \left(q\_m \cdot q\_m\right)}{\left(\left|r\right| + \left|p\right|\right) + r} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow q_m 2.0))))
   (if (<= t_0 5e-71)
     (* (- p) (- (* (/ (+ (fabs p) (- (fabs r) r)) p) -0.5) 0.5))
     (if (<= t_0 5e+171)
       (* (/ (* -4.0 (* q_m q_m)) (+ (+ (fabs r) (fabs p)) r)) 0.5)
       (- q_m)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = 4.0 * pow(q_m, 2.0);
	double tmp;
	if (t_0 <= 5e-71) {
		tmp = -p * ((((fabs(p) + (fabs(r) - r)) / p) * -0.5) - 0.5);
	} else if (t_0 <= 5e+171) {
		tmp = ((-4.0 * (q_m * q_m)) / ((fabs(r) + fabs(p)) + r)) * 0.5;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (q_m ** 2.0d0)
    if (t_0 <= 5d-71) then
        tmp = -p * ((((abs(p) + (abs(r) - r)) / p) * (-0.5d0)) - 0.5d0)
    else if (t_0 <= 5d+171) then
        tmp = (((-4.0d0) * (q_m * q_m)) / ((abs(r) + abs(p)) + r)) * 0.5d0
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double t_0 = 4.0 * Math.pow(q_m, 2.0);
	double tmp;
	if (t_0 <= 5e-71) {
		tmp = -p * ((((Math.abs(p) + (Math.abs(r) - r)) / p) * -0.5) - 0.5);
	} else if (t_0 <= 5e+171) {
		tmp = ((-4.0 * (q_m * q_m)) / ((Math.abs(r) + Math.abs(p)) + r)) * 0.5;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	t_0 = 4.0 * math.pow(q_m, 2.0)
	tmp = 0
	if t_0 <= 5e-71:
		tmp = -p * ((((math.fabs(p) + (math.fabs(r) - r)) / p) * -0.5) - 0.5)
	elif t_0 <= 5e+171:
		tmp = ((-4.0 * (q_m * q_m)) / ((math.fabs(r) + math.fabs(p)) + r)) * 0.5
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(4.0 * (q_m ^ 2.0))
	tmp = 0.0
	if (t_0 <= 5e-71)
		tmp = Float64(Float64(-p) * Float64(Float64(Float64(Float64(abs(p) + Float64(abs(r) - r)) / p) * -0.5) - 0.5));
	elseif (t_0 <= 5e+171)
		tmp = Float64(Float64(Float64(-4.0 * Float64(q_m * q_m)) / Float64(Float64(abs(r) + abs(p)) + r)) * 0.5);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	t_0 = 4.0 * (q_m ^ 2.0);
	tmp = 0.0;
	if (t_0 <= 5e-71)
		tmp = -p * ((((abs(p) + (abs(r) - r)) / p) * -0.5) - 0.5);
	elseif (t_0 <= 5e+171)
		tmp = ((-4.0 * (q_m * q_m)) / ((abs(r) + abs(p)) + r)) * 0.5;
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-71], N[((-p) * N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+171], N[(N[(N[(-4.0 * N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 4.99999999999999998e-71

   1. Initial program 24.3%

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

   3. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 61.1%

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

   if 4.99999999999999998e-71 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 5.0000000000000004e171

   1. Initial program 31.7%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\left(\left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 30.1%

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

   8. Taylor expanded in q around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 46.5

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

   10. Applied rewrites 46.5%

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

   11. Taylor expanded in r around inf

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

   13. Applied rewrites 32.1%

       \[\leadsto \frac{-4 \cdot \left(q \cdot q\right)}{\left(\left|r\right| + \left|p\right|\right) + r} \cdot 0.5 \]

   if 5.0000000000000004e171 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 20.8%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(q\right) \]

      2. lower-neg.f64 68.0

         \[\leadsto -q \]

   5. Applied rewrites 68.0%

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

Alternative 4: 58.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;\left(-p\right) \cdot \left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p} \cdot -0.5 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 4.8e-12)
   (* (- p) (- (* (/ (+ (fabs p) (- (fabs r) r)) p) -0.5) 0.5))
   (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 4.8e-12) {
		tmp = -p * ((((fabs(p) + (fabs(r) - r)) / p) * -0.5) - 0.5);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 4.8d-12) then
        tmp = -p * ((((abs(p) + (abs(r) - r)) / p) * (-0.5d0)) - 0.5d0)
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 4.8e-12) {
		tmp = -p * ((((Math.abs(p) + (Math.abs(r) - r)) / p) * -0.5) - 0.5);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 4.8e-12:
		tmp = -p * ((((math.fabs(p) + (math.fabs(r) - r)) / p) * -0.5) - 0.5)
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 4.8e-12)
		tmp = Float64(Float64(-p) * Float64(Float64(Float64(Float64(abs(p) + Float64(abs(r) - r)) / p) * -0.5) - 0.5));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 4.8e-12)
		tmp = -p * ((((abs(p) + (abs(r) - r)) / p) * -0.5) - 0.5);
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 4.8e-12], N[((-p) * N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 4.79999999999999974e-12

   1. Initial program 24.2%

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

   3. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 58.4%

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

   if 4.79999999999999974e-12 < q

   1. Initial program 24.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(q\right) \]

      2. lower-neg.f64 57.6

         \[\leadsto -q \]

   5. Applied rewrites 57.6%

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

Alternative 5: 58.0% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left(\left|r\right| - r\right) + \left|p\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 4.8e-12) (* 0.5 (+ p (+ (- (fabs r) 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 (q_m <= 4.8e-12) {
		tmp = 0.5 * (p + ((fabs(r) - r) + fabs(p)));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 4.8d-12) then
        tmp = 0.5d0 * (p + ((abs(r) - r) + abs(p)))
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 4.8e-12) {
		tmp = 0.5 * (p + ((Math.abs(r) - r) + Math.abs(p)));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 4.8e-12:
		tmp = 0.5 * (p + ((math.fabs(r) - r) + math.fabs(p)))
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 4.8e-12)
		tmp = Float64(0.5 * Float64(p + Float64(Float64(abs(r) - r) + abs(p))));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 4.8e-12)
		tmp = 0.5 * (p + ((abs(r) - r) + abs(p)));
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 4.8e-12], N[(0.5 * N[(p + N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 4.79999999999999974e-12

   1. Initial program 24.2%

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

   3. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+r- N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-fabs.f64 58.4

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

   8. Applied rewrites 58.4%

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

   if 4.79999999999999974e-12 < q

   1. Initial program 24.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(q\right) \]

      2. lower-neg.f64 57.6

         \[\leadsto -q \]

   5. Applied rewrites 57.6%

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

Alternative 6: 40.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 7.5 \cdot 10^{-75}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) - r\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 7.5e-75) (* 0.5 (- (+ (fabs p) (fabs r)) r)) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 7.5e-75) {
		tmp = 0.5 * ((fabs(p) + fabs(r)) - r);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 7.5d-75) then
        tmp = 0.5d0 * ((abs(p) + abs(r)) - r)
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 7.5e-75) {
		tmp = 0.5 * ((Math.abs(p) + Math.abs(r)) - r);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 7.5e-75:
		tmp = 0.5 * ((math.fabs(p) + math.fabs(r)) - r)
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 7.5e-75)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + abs(r)) - r));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 7.5e-75)
		tmp = 0.5 * ((abs(p) + abs(r)) - r);
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 7.5e-75], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 7.50000000000000017e-75

   1. Initial program 24.7%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 21.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 21.9

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

   7. Applied rewrites 21.9%

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

   if 7.50000000000000017e-75 < q

   1. Initial program 24.3%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(q\right) \]

      2. lower-neg.f64 51.8

         \[\leadsto -q \]

   5. Applied rewrites 51.8%

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

Alternative 7: 35.5% accurate, 83.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(-q_m)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in q around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(q\right) \]

   2. lower-neg.f64 35.5

      \[\leadsto -q \]

5. Applied rewrites 35.5%

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

Alternative 8: 3.3% accurate, 250.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in q around -inf

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

5. Applied rewrites 3.3%

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

Reproduce

herbie shell --seed 2025086 
(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)))))))