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

Percentage Accurate: 23.6% → 60.8%

Time: 6.9s

Alternatives: 8

Speedup: 19.7×

Specification

Math

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

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: 23.6% accurate, 1.0× speedup

Math

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

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: 60.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_2 := t\_1 - \mathsf{max}\left(p, r\right)\\ t_3 := \mathsf{max}\left(p, r\right) - \mathsf{min}\left(p, r\right)\\ \mathbf{if}\;\left|q\right| \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{min}\left(p, r\right) \cdot \left(0.5 + 0.5 \cdot \frac{t\_0 - \left(\mathsf{max}\left(p, r\right) - t\_1\right)}{\mathsf{min}\left(p, r\right)}\right)\\ \mathbf{elif}\;\left|q\right| \leq 2.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{-1 \cdot \left(\mathsf{min}\left(p, r\right) \cdot \mathsf{fma}\left(-2, t\_2, -1 \cdot \frac{\mathsf{fma}\left(4, {\left(\left|q\right|\right)}^{2}, {t\_2}^{2}\right)}{\mathsf{min}\left(p, r\right)}\right)\right)}{\left(t\_1 - \sqrt{\mathsf{fma}\left(\left|q\right| \cdot 4, \left|q\right|, t\_3 \cdot t\_3\right)}\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r)))
        (t_1 (fabs (fmax p r)))
        (t_2 (- t_1 (fmax p r)))
        (t_3 (- (fmax p r) (fmin p r))))
   (if (<= (fabs q) 2.6e-128)
     (* (fmin p r) (+ 0.5 (* 0.5 (/ (- t_0 (- (fmax p r) t_1)) (fmin p r)))))
     (if (<= (fabs q) 2.6e+34)
       (*
        (/ 1.0 2.0)
        (/
         (*
          -1.0
          (*
           (fmin p r)
           (fma
            -2.0
            t_2
            (*
             -1.0
             (/ (fma 4.0 (pow (fabs q) 2.0) (pow t_2 2.0)) (fmin p r))))))
         (- (- t_1 (sqrt (fma (* (fabs q) 4.0) (fabs q) (* t_3 t_3)))) t_0)))
       (- (fabs q))))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double t_2 = t_1 - fmax(p, r);
	double t_3 = fmax(p, r) - fmin(p, r);
	double tmp;
	if (fabs(q) <= 2.6e-128) {
		tmp = fmin(p, r) * (0.5 + (0.5 * ((t_0 - (fmax(p, r) - t_1)) / fmin(p, r))));
	} else if (fabs(q) <= 2.6e+34) {
		tmp = (1.0 / 2.0) * ((-1.0 * (fmin(p, r) * fma(-2.0, t_2, (-1.0 * (fma(4.0, pow(fabs(q), 2.0), pow(t_2, 2.0)) / fmin(p, r)))))) / ((t_1 - sqrt(fma((fabs(q) * 4.0), fabs(q), (t_3 * t_3)))) - t_0));
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	t_2 = Float64(t_1 - fmax(p, r))
	t_3 = Float64(fmax(p, r) - fmin(p, r))
	tmp = 0.0
	if (abs(q) <= 2.6e-128)
		tmp = Float64(fmin(p, r) * Float64(0.5 + Float64(0.5 * Float64(Float64(t_0 - Float64(fmax(p, r) - t_1)) / fmin(p, r)))));
	elseif (abs(q) <= 2.6e+34)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(-1.0 * Float64(fmin(p, r) * fma(-2.0, t_2, Float64(-1.0 * Float64(fma(4.0, (abs(q) ^ 2.0), (t_2 ^ 2.0)) / fmin(p, r)))))) / Float64(Float64(t_1 - sqrt(fma(Float64(abs(q) * 4.0), abs(q), Float64(t_3 * t_3)))) - t_0)));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Max[p, r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Max[p, r], $MachinePrecision] - N[Min[p, r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 2.6e-128], N[(N[Min[p, r], $MachinePrecision] * N[(0.5 + N[(0.5 * N[(N[(t$95$0 - N[(N[Max[p, r], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[q], $MachinePrecision], 2.6e+34], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(-1.0 * N[(N[Min[p, r], $MachinePrecision] * N[(-2.0 * t$95$2 + N[(-1.0 * N[(N[(4.0 * N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 - N[Sqrt[N[(N[(N[Abs[q], $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[q], $MachinePrecision] + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 2.5999999999999998e-128

   1. Initial program 23.6%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 8.4%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(-1 \cdot \frac{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - r, \frac{{q}^{2}}{p}\right)}{p} - 0.5\right)\right)} \]

   5. Taylor expanded in p around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-fabs.f64 8.0%

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

   7. Applied rewrites 8.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 16.4%

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

   9. Applied rewrites 16.4%

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

   if 2.5999999999999998e-128 < q < 2.6e34

   1. Initial program 23.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 20.3%

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

   4. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 26.5%

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

   if 2.6e34 < q

   1. Initial program 23.6%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.6%

         \[\leadsto -1 \cdot \color{blue}{q} \]

   4. Applied rewrites 19.6%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{q} \]

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.6%

         \[\leadsto -q \]

   6. Applied rewrites 19.6%

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

Alternative 2: 60.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_2 := \mathsf{max}\left(p, r\right) - t\_1\\ t_3 := \mathsf{min}\left(p, r\right) - \mathsf{max}\left(p, r\right)\\ t_4 := t\_1 - \mathsf{max}\left(p, r\right)\\ \mathbf{if}\;\left|q\right| \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{min}\left(p, r\right) \cdot \left(0.5 + 0.5 \cdot \frac{t\_0 - t\_2}{\mathsf{min}\left(p, r\right)}\right)\\ \mathbf{elif}\;\left|q\right| \leq 1900000000:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(-\mathsf{min}\left(p, r\right)\right) \cdot \mathsf{fma}\left(t\_4, -2, \frac{\mathsf{fma}\left(t\_2, t\_4, -4 \cdot \left(\left|q\right| \cdot \left|q\right|\right)\right)}{\mathsf{min}\left(p, r\right)}\right)\right)}{t\_1 - \left(t\_0 + \sqrt{\mathsf{fma}\left(t\_3, t\_3, \left(4 \cdot \left|q\right|\right) \cdot \left|q\right|\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r)))
        (t_1 (fabs (fmax p r)))
        (t_2 (- (fmax p r) t_1))
        (t_3 (- (fmin p r) (fmax p r)))
        (t_4 (- t_1 (fmax p r))))
   (if (<= (fabs q) 2.6e-128)
     (* (fmin p r) (+ 0.5 (* 0.5 (/ (- t_0 t_2) (fmin p r)))))
     (if (<= (fabs q) 1900000000.0)
       (/
        (*
         0.5
         (*
          (- (fmin p r))
          (fma
           t_4
           -2.0
           (/ (fma t_2 t_4 (* -4.0 (* (fabs q) (fabs q)))) (fmin p r)))))
        (- t_1 (+ t_0 (sqrt (fma t_3 t_3 (* (* 4.0 (fabs q)) (fabs q)))))))
       (- (fabs q))))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double t_2 = fmax(p, r) - t_1;
	double t_3 = fmin(p, r) - fmax(p, r);
	double t_4 = t_1 - fmax(p, r);
	double tmp;
	if (fabs(q) <= 2.6e-128) {
		tmp = fmin(p, r) * (0.5 + (0.5 * ((t_0 - t_2) / fmin(p, r))));
	} else if (fabs(q) <= 1900000000.0) {
		tmp = (0.5 * (-fmin(p, r) * fma(t_4, -2.0, (fma(t_2, t_4, (-4.0 * (fabs(q) * fabs(q)))) / fmin(p, r))))) / (t_1 - (t_0 + sqrt(fma(t_3, t_3, ((4.0 * fabs(q)) * fabs(q))))));
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	t_2 = Float64(fmax(p, r) - t_1)
	t_3 = Float64(fmin(p, r) - fmax(p, r))
	t_4 = Float64(t_1 - fmax(p, r))
	tmp = 0.0
	if (abs(q) <= 2.6e-128)
		tmp = Float64(fmin(p, r) * Float64(0.5 + Float64(0.5 * Float64(Float64(t_0 - t_2) / fmin(p, r)))));
	elseif (abs(q) <= 1900000000.0)
		tmp = Float64(Float64(0.5 * Float64(Float64(-fmin(p, r)) * fma(t_4, -2.0, Float64(fma(t_2, t_4, Float64(-4.0 * Float64(abs(q) * abs(q)))) / fmin(p, r))))) / Float64(t_1 - Float64(t_0 + sqrt(fma(t_3, t_3, Float64(Float64(4.0 * abs(q)) * abs(q)))))));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Max[p, r], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Min[p, r], $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 - N[Max[p, r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 2.6e-128], N[(N[Min[p, r], $MachinePrecision] * N[(0.5 + N[(0.5 * N[(N[(t$95$0 - t$95$2), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[q], $MachinePrecision], 1900000000.0], N[(N[(0.5 * N[((-N[Min[p, r], $MachinePrecision]) * N[(t$95$4 * -2.0 + N[(N[(t$95$2 * t$95$4 + N[(-4.0 * N[(N[Abs[q], $MachinePrecision] * N[Abs[q], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(t$95$0 + N[Sqrt[N[(t$95$3 * t$95$3 + N[(N[(4.0 * N[Abs[q], $MachinePrecision]), $MachinePrecision] * N[Abs[q], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]]]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 2.5999999999999998e-128

   1. Initial program 23.6%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 8.4%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(-1 \cdot \frac{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - r, \frac{{q}^{2}}{p}\right)}{p} - 0.5\right)\right)} \]

   5. Taylor expanded in p around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-fabs.f64 8.0%

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

   7. Applied rewrites 8.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 16.4%

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

   9. Applied rewrites 16.4%

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

   if 2.5999999999999998e-128 < q < 1.9e9

   1. Initial program 23.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 20.3%

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

   4. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 26.5%

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

   8. Applied rewrites 25.0%

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

   if 1.9e9 < q

   1. Initial program 23.6%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.6%

         \[\leadsto -1 \cdot \color{blue}{q} \]

   4. Applied rewrites 19.6%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{q} \]

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.6%

         \[\leadsto -q \]

   6. Applied rewrites 19.6%

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

Alternative 3: 60.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_2 := \mathsf{max}\left(p, r\right) - t\_1\\ t_3 := \mathsf{max}\left(p, r\right) - \mathsf{min}\left(p, r\right)\\ t_4 := t\_1 - \mathsf{max}\left(p, r\right)\\ \mathbf{if}\;\left|q\right| \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{min}\left(p, r\right) \cdot \left(0.5 + 0.5 \cdot \frac{t\_0 - t\_2}{\mathsf{min}\left(p, r\right)}\right)\\ \mathbf{elif}\;\left|q\right| \leq 1900000000:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, t\_4, \frac{\mathsf{fma}\left(t\_2, t\_4, \left(\left|q\right| \cdot \left|q\right|\right) \cdot -4\right)}{\mathsf{min}\left(p, r\right)}\right) \cdot \left(-\mathsf{min}\left(p, r\right)\right)\right) \cdot \frac{-0.5}{\sqrt{\mathsf{fma}\left(t\_3, t\_3, \left(4 \cdot \left|q\right|\right) \cdot \left|q\right|\right)} - \left(t\_1 - t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r)))
        (t_1 (fabs (fmax p r)))
        (t_2 (- (fmax p r) t_1))
        (t_3 (- (fmax p r) (fmin p r)))
        (t_4 (- t_1 (fmax p r))))
   (if (<= (fabs q) 2.6e-128)
     (* (fmin p r) (+ 0.5 (* 0.5 (/ (- t_0 t_2) (fmin p r)))))
     (if (<= (fabs q) 1900000000.0)
       (*
        (*
         (fma
          -2.0
          t_4
          (/ (fma t_2 t_4 (* (* (fabs q) (fabs q)) -4.0)) (fmin p r)))
         (- (fmin p r)))
        (/
         -0.5
         (- (sqrt (fma t_3 t_3 (* (* 4.0 (fabs q)) (fabs q)))) (- t_1 t_0))))
       (- (fabs q))))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double t_2 = fmax(p, r) - t_1;
	double t_3 = fmax(p, r) - fmin(p, r);
	double t_4 = t_1 - fmax(p, r);
	double tmp;
	if (fabs(q) <= 2.6e-128) {
		tmp = fmin(p, r) * (0.5 + (0.5 * ((t_0 - t_2) / fmin(p, r))));
	} else if (fabs(q) <= 1900000000.0) {
		tmp = (fma(-2.0, t_4, (fma(t_2, t_4, ((fabs(q) * fabs(q)) * -4.0)) / fmin(p, r))) * -fmin(p, r)) * (-0.5 / (sqrt(fma(t_3, t_3, ((4.0 * fabs(q)) * fabs(q)))) - (t_1 - t_0)));
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	t_2 = Float64(fmax(p, r) - t_1)
	t_3 = Float64(fmax(p, r) - fmin(p, r))
	t_4 = Float64(t_1 - fmax(p, r))
	tmp = 0.0
	if (abs(q) <= 2.6e-128)
		tmp = Float64(fmin(p, r) * Float64(0.5 + Float64(0.5 * Float64(Float64(t_0 - t_2) / fmin(p, r)))));
	elseif (abs(q) <= 1900000000.0)
		tmp = Float64(Float64(fma(-2.0, t_4, Float64(fma(t_2, t_4, Float64(Float64(abs(q) * abs(q)) * -4.0)) / fmin(p, r))) * Float64(-fmin(p, r))) * Float64(-0.5 / Float64(sqrt(fma(t_3, t_3, Float64(Float64(4.0 * abs(q)) * abs(q)))) - Float64(t_1 - t_0))));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Max[p, r], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Max[p, r], $MachinePrecision] - N[Min[p, r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 - N[Max[p, r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 2.6e-128], N[(N[Min[p, r], $MachinePrecision] * N[(0.5 + N[(0.5 * N[(N[(t$95$0 - t$95$2), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[q], $MachinePrecision], 1900000000.0], N[(N[(N[(-2.0 * t$95$4 + N[(N[(t$95$2 * t$95$4 + N[(N[(N[Abs[q], $MachinePrecision] * N[Abs[q], $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Min[p, r], $MachinePrecision])), $MachinePrecision] * N[(-0.5 / N[(N[Sqrt[N[(t$95$3 * t$95$3 + N[(N[(4.0 * N[Abs[q], $MachinePrecision]), $MachinePrecision] * N[Abs[q], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]]]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 2.5999999999999998e-128

   1. Initial program 23.6%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 8.4%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(-1 \cdot \frac{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - r, \frac{{q}^{2}}{p}\right)}{p} - 0.5\right)\right)} \]

   5. Taylor expanded in p around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-fabs.f64 8.0%

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

   7. Applied rewrites 8.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 16.4%

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

   9. Applied rewrites 16.4%

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

   if 2.5999999999999998e-128 < q < 1.9e9

   1. Initial program 23.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 20.3%

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

   4. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 26.5%

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

   8. Applied rewrites 25.0%

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

   10. Applied rewrites 25.0%

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

   if 1.9e9 < q

   1. Initial program 23.6%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.6%

         \[\leadsto -1 \cdot \color{blue}{q} \]

   4. Applied rewrites 19.6%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{q} \]

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.6%

         \[\leadsto -q \]

   6. Applied rewrites 19.6%

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

Alternative 4: 59.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_2 := \mathsf{max}\left(p, r\right) - \mathsf{min}\left(p, r\right)\\ \mathbf{if}\;\left|q\right| \leq 2.65 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{min}\left(p, r\right) \cdot \left(0.5 + 0.5 \cdot \frac{t\_0 - \left(\mathsf{max}\left(p, r\right) - t\_1\right)}{\mathsf{min}\left(p, r\right)}\right)\\ \mathbf{elif}\;\left|q\right| \leq 3.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot {\left(\left|q\right|\right)}^{2}}{\left(t\_1 - \sqrt{\mathsf{fma}\left(\left|q\right| \cdot 4, \left|q\right|, t\_2 \cdot t\_2\right)}\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(t\_0 + t\_1\right) - 2 \cdot \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r)))
        (t_1 (fabs (fmax p r)))
        (t_2 (- (fmax p r) (fmin p r))))
   (if (<= (fabs q) 2.65e-106)
     (* (fmin p r) (+ 0.5 (* 0.5 (/ (- t_0 (- (fmax p r) t_1)) (fmin p r)))))
     (if (<= (fabs q) 3.6e+147)
       (*
        (/ 1.0 2.0)
        (/
         (* 4.0 (pow (fabs q) 2.0))
         (- (- t_1 (sqrt (fma (* (fabs q) 4.0) (fabs q) (* t_2 t_2)))) t_0)))
       (* (/ 1.0 2.0) (- (+ t_0 t_1) (* 2.0 (fabs q))))))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double t_2 = fmax(p, r) - fmin(p, r);
	double tmp;
	if (fabs(q) <= 2.65e-106) {
		tmp = fmin(p, r) * (0.5 + (0.5 * ((t_0 - (fmax(p, r) - t_1)) / fmin(p, r))));
	} else if (fabs(q) <= 3.6e+147) {
		tmp = (1.0 / 2.0) * ((4.0 * pow(fabs(q), 2.0)) / ((t_1 - sqrt(fma((fabs(q) * 4.0), fabs(q), (t_2 * t_2)))) - t_0));
	} else {
		tmp = (1.0 / 2.0) * ((t_0 + t_1) - (2.0 * fabs(q)));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	t_2 = Float64(fmax(p, r) - fmin(p, r))
	tmp = 0.0
	if (abs(q) <= 2.65e-106)
		tmp = Float64(fmin(p, r) * Float64(0.5 + Float64(0.5 * Float64(Float64(t_0 - Float64(fmax(p, r) - t_1)) / fmin(p, r)))));
	elseif (abs(q) <= 3.6e+147)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(4.0 * (abs(q) ^ 2.0)) / Float64(Float64(t_1 - sqrt(fma(Float64(abs(q) * 4.0), abs(q), Float64(t_2 * t_2)))) - t_0)));
	else
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(t_0 + t_1) - Float64(2.0 * abs(q))));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Max[p, r], $MachinePrecision] - N[Min[p, r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 2.65e-106], N[(N[Min[p, r], $MachinePrecision] * N[(0.5 + N[(0.5 * N[(N[(t$95$0 - N[(N[Max[p, r], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[q], $MachinePrecision], 3.6e+147], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(4.0 * N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 - N[Sqrt[N[(N[(N[Abs[q], $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[q], $MachinePrecision] + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(t$95$0 + t$95$1), $MachinePrecision] - N[(2.0 * N[Abs[q], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 2.6499999999999999e-106

   1. Initial program 23.6%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 8.4%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(-1 \cdot \frac{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - r, \frac{{q}^{2}}{p}\right)}{p} - 0.5\right)\right)} \]

   5. Taylor expanded in p around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-fabs.f64 8.0%

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

   7. Applied rewrites 8.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 16.4%

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

   9. Applied rewrites 16.4%

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

   if 2.6499999999999999e-106 < q < 3.6000000000000002e147

   1. Initial program 23.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 20.3%

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

   4. Taylor expanded in q around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 35.1%

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

   6. Applied rewrites 35.1%

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

   if 3.6000000000000002e147 < q

   1. Initial program 23.6%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 18.6%

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

   4. Applied rewrites 18.6%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \color{blue}{2 \cdot q}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 59.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(p, r\right) - \mathsf{max}\left(p, r\right)\\ t_1 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_2 := \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;\left|q\right| \leq 2.65 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{min}\left(p, r\right) \cdot \left(0.5 + 0.5 \cdot \frac{t\_1 - \left(\mathsf{max}\left(p, r\right) - t\_2\right)}{\mathsf{min}\left(p, r\right)}\right)\\ \mathbf{elif}\;\left|q\right| \leq 3.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{2 \cdot {\left(\left|q\right|\right)}^{2}}{t\_2 - \left(t\_1 + \sqrt{\mathsf{fma}\left(t\_0, t\_0, \left(4 \cdot \left|q\right|\right) \cdot \left|q\right|\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(t\_1 + t\_2\right) - 2 \cdot \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (- (fmin p r) (fmax p r)))
        (t_1 (fabs (fmin p r)))
        (t_2 (fabs (fmax p r))))
   (if (<= (fabs q) 2.65e-106)
     (* (fmin p r) (+ 0.5 (* 0.5 (/ (- t_1 (- (fmax p r) t_2)) (fmin p r)))))
     (if (<= (fabs q) 3.6e+147)
       (/
        (* 2.0 (pow (fabs q) 2.0))
        (- t_2 (+ t_1 (sqrt (fma t_0 t_0 (* (* 4.0 (fabs q)) (fabs q)))))))
       (* (/ 1.0 2.0) (- (+ t_1 t_2) (* 2.0 (fabs q))))))))

C

double code(double p, double r, double q) {
	double t_0 = fmin(p, r) - fmax(p, r);
	double t_1 = fabs(fmin(p, r));
	double t_2 = fabs(fmax(p, r));
	double tmp;
	if (fabs(q) <= 2.65e-106) {
		tmp = fmin(p, r) * (0.5 + (0.5 * ((t_1 - (fmax(p, r) - t_2)) / fmin(p, r))));
	} else if (fabs(q) <= 3.6e+147) {
		tmp = (2.0 * pow(fabs(q), 2.0)) / (t_2 - (t_1 + sqrt(fma(t_0, t_0, ((4.0 * fabs(q)) * fabs(q))))));
	} else {
		tmp = (1.0 / 2.0) * ((t_1 + t_2) - (2.0 * fabs(q)));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = Float64(fmin(p, r) - fmax(p, r))
	t_1 = abs(fmin(p, r))
	t_2 = abs(fmax(p, r))
	tmp = 0.0
	if (abs(q) <= 2.65e-106)
		tmp = Float64(fmin(p, r) * Float64(0.5 + Float64(0.5 * Float64(Float64(t_1 - Float64(fmax(p, r) - t_2)) / fmin(p, r)))));
	elseif (abs(q) <= 3.6e+147)
		tmp = Float64(Float64(2.0 * (abs(q) ^ 2.0)) / Float64(t_2 - Float64(t_1 + sqrt(fma(t_0, t_0, Float64(Float64(4.0 * abs(q)) * abs(q)))))));
	else
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(t_1 + t_2) - Float64(2.0 * abs(q))));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[(N[Min[p, r], $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 2.65e-106], N[(N[Min[p, r], $MachinePrecision] * N[(0.5 + N[(0.5 * N[(N[(t$95$1 - N[(N[Max[p, r], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[q], $MachinePrecision], 3.6e+147], N[(N[(2.0 * N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - N[(t$95$1 + N[Sqrt[N[(t$95$0 * t$95$0 + N[(N[(4.0 * N[Abs[q], $MachinePrecision]), $MachinePrecision] * N[Abs[q], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(t$95$1 + t$95$2), $MachinePrecision] - N[(2.0 * N[Abs[q], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 2.6499999999999999e-106

   1. Initial program 23.6%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 8.4%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(-1 \cdot \frac{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - r, \frac{{q}^{2}}{p}\right)}{p} - 0.5\right)\right)} \]

   5. Taylor expanded in p around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-fabs.f64 8.0%

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

   7. Applied rewrites 8.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 16.4%

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

   9. Applied rewrites 16.4%

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

   if 2.6499999999999999e-106 < q < 3.6000000000000002e147

   1. Initial program 23.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 20.3%

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

   4. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 26.5%

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

   8. Applied rewrites 25.0%

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

   9. Taylor expanded in q around inf

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 32.2%

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

   11. Applied rewrites 32.2%

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

   if 3.6000000000000002e147 < q

   1. Initial program 23.6%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 18.6%

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

   4. Applied rewrites 18.6%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \color{blue}{2 \cdot q}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 57.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 2.5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{min}\left(p, r\right) \cdot \left(0.5 + 0.5 \cdot \frac{\left|\mathsf{min}\left(p, r\right)\right| - \left(\mathsf{max}\left(p, r\right) - \left|\mathsf{max}\left(p, r\right)\right|\right)}{\mathsf{min}\left(p, r\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 2.5e-19)
   (*
    (fmin p r)
    (+
     0.5
     (*
      0.5
      (/ (- (fabs (fmin p r)) (- (fmax p r) (fabs (fmax p r)))) (fmin p r)))))
   (- (fabs q))))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 2.5e-19) {
		tmp = fmin(p, r) * (0.5 + (0.5 * ((fabs(fmin(p, r)) - (fmax(p, r) - fabs(fmax(p, r)))) / fmin(p, r))));
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(q) <= 2.5d-19) then
        tmp = fmin(p, r) * (0.5d0 + (0.5d0 * ((abs(fmin(p, r)) - (fmax(p, r) - abs(fmax(p, r)))) / fmin(p, r))))
    else
        tmp = -abs(q)
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (Math.abs(q) <= 2.5e-19) {
		tmp = fmin(p, r) * (0.5 + (0.5 * ((Math.abs(fmin(p, r)) - (fmax(p, r) - Math.abs(fmax(p, r)))) / fmin(p, r))));
	} else {
		tmp = -Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 2.5e-19:
		tmp = fmin(p, r) * (0.5 + (0.5 * ((math.fabs(fmin(p, r)) - (fmax(p, r) - math.fabs(fmax(p, r)))) / fmin(p, r))))
	else:
		tmp = -math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 2.5e-19)
		tmp = Float64(fmin(p, r) * Float64(0.5 + Float64(0.5 * Float64(Float64(abs(fmin(p, r)) - Float64(fmax(p, r) - abs(fmax(p, r)))) / fmin(p, r)))));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 2.5e-19)
		tmp = min(p, r) * (0.5 + (0.5 * ((abs(min(p, r)) - (max(p, r) - abs(max(p, r)))) / min(p, r))));
	else
		tmp = -abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 2.5e-19], N[(N[Min[p, r], $MachinePrecision] * N[(0.5 + N[(0.5 * N[(N[(N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision] - N[(N[Max[p, r], $MachinePrecision] - N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if q < 2.5000000000000002e-19

   1. Initial program 23.6%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 8.4%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(-1 \cdot \frac{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - r, \frac{{q}^{2}}{p}\right)}{p} - 0.5\right)\right)} \]

   5. Taylor expanded in p around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-fabs.f64 8.0%

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

   7. Applied rewrites 8.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 16.4%

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

   9. Applied rewrites 16.4%

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

   if 2.5000000000000002e-19 < q

   1. Initial program 23.6%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.6%

         \[\leadsto -1 \cdot \color{blue}{q} \]

   4. Applied rewrites 19.6%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{q} \]

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.6%

         \[\leadsto -q \]

   6. Applied rewrites 19.6%

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

Alternative 7: 49.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 4.2 \cdot 10^{-62}:\\ \;\;\;\;\left|\mathsf{max}\left(p, r\right)\right| - \mathsf{max}\left(p, r\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 4.2e-62) (- (fabs (fmax p r)) (fmax p r)) (- (fabs q))))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 4.2e-62) {
		tmp = fabs(fmax(p, r)) - fmax(p, r);
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(q) <= 4.2d-62) then
        tmp = abs(fmax(p, r)) - fmax(p, r)
    else
        tmp = -abs(q)
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (Math.abs(q) <= 4.2e-62) {
		tmp = Math.abs(fmax(p, r)) - fmax(p, r);
	} else {
		tmp = -Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 4.2e-62:
		tmp = math.fabs(fmax(p, r)) - fmax(p, r)
	else:
		tmp = -math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 4.2e-62)
		tmp = Float64(abs(fmax(p, r)) - fmax(p, r));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 4.2e-62)
		tmp = abs(max(p, r)) - max(p, r);
	else
		tmp = -abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 4.2e-62], N[(N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if q < 4.1999999999999998e-62

   1. Initial program 23.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 20.3%

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

   4. Taylor expanded in p around -inf

      \[\leadsto \color{blue}{\left|r\right| - r} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left|r\right| - \color{blue}{r} \]

      2. lower-fabs.f64 11.6%

         \[\leadsto \left|r\right| - r \]

   6. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\left|r\right| - r} \]

   if 4.1999999999999998e-62 < q

   1. Initial program 23.6%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.6%

         \[\leadsto -1 \cdot \color{blue}{q} \]

   4. Applied rewrites 19.6%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{q} \]

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.6%

         \[\leadsto -q \]

   6. Applied rewrites 19.6%

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

Alternative 8: 35.8% accurate, 19.7× speedup

Math

\[-\left|q\right| \]

FPCore

(FPCore (p r q) :precision binary64 (- (fabs q)))

C

double code(double p, double r, double q) {
	return -fabs(q);
}

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 = -abs(q)
end function

Java

public static double code(double p, double r, double q) {
	return -Math.abs(q);
}

Python

def code(p, r, q):
	return -math.fabs(q)

Julia

function code(p, r, q)
	return Float64(-abs(q))
end

MATLAB

function tmp = code(p, r, q)
	tmp = -abs(q);
end

Wolfram

code[p_, r_, q_] := (-N[Abs[q], $MachinePrecision])

Derivation

1. Initial program 23.6%

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

2. Taylor expanded in q around inf

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

   1. lower-*.f64 19.6%

      \[\leadsto -1 \cdot \color{blue}{q} \]

4. Applied rewrites 19.6%

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

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{q} \]

   2. mul-1-neg N/A

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

   3. lower-neg.f64 19.6%

      \[\leadsto -q \]

6. Applied rewrites 19.6%

   \[\leadsto -q \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025196 
(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)))))))