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

Percentage Accurate: 23.9% → 65.2%

Time: 7.6s

Alternatives: 8

Speedup: 4.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 23.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|p - r\right|\\ \mathbf{if}\;r \leq -3.9 \cdot 10^{-236}:\\ \;\;\;\;\mathsf{fma}\left(\frac{q}{p}, q, \left(\left|r\right| - r\right) \cdot 0.5\right)\\ \mathbf{elif}\;r \leq 2.9 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \mathsf{hypot}\left(r - p, q + q\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|r\right| - p\right) - t\_0, 0.5, \left(-q\right) \cdot \frac{q}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (- p r))))
   (if (<= r -3.9e-236)
     (fma (/ q p) q (* (- (fabs r) r) 0.5))
     (if (<= r 2.9e+124)
       (* 0.5 (- (+ (fabs p) (fabs r)) (hypot (- r p) (+ q q))))
       (fma (- (- (fabs r) p) t_0) 0.5 (* (- q) (/ q t_0)))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs((p - r));
	double tmp;
	if (r <= -3.9e-236) {
		tmp = fma((q / p), q, ((fabs(r) - r) * 0.5));
	} else if (r <= 2.9e+124) {
		tmp = 0.5 * ((fabs(p) + fabs(r)) - hypot((r - p), (q + q)));
	} else {
		tmp = fma(((fabs(r) - p) - t_0), 0.5, (-q * (q / t_0)));
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = abs(Float64(p - r))
	tmp = 0.0
	if (r <= -3.9e-236)
		tmp = fma(Float64(q / p), q, Float64(Float64(abs(r) - r) * 0.5));
	elseif (r <= 2.9e+124)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + abs(r)) - hypot(Float64(r - p), Float64(q + q))));
	else
		tmp = fma(Float64(Float64(abs(r) - p) - t_0), 0.5, Float64(Float64(-q) * Float64(q / t_0)));
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[(p - r), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[r, -3.9e-236], N[(N[(q / p), $MachinePrecision] * q + N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 2.9e+124], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(r - p), $MachinePrecision] ^ 2 + N[(q + q), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision] - t$95$0), $MachinePrecision] * 0.5 + N[((-q) * N[(q / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < -3.9e-236

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 38.5

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

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|r\right| - r, \frac{{q}^{2}}{p}\right)} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. mult-flip N/A

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

       5. lift-pow.f64 N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. mult-flip-rev N/A

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

       11. lower-/.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 39.9

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

   11. Applied rewrites 39.9%

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

   if -3.9e-236 < r < 2.90000000000000021e124

   1. Initial program 23.9%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. sub-flip N/A

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

      12. add-flip N/A

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

      13. remove-double-neg N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.3

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

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \mathsf{hypot}\left(r - p, q \cdot 2\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 52.3

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

   7. Applied rewrites 52.3%

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

   if 2.90000000000000021e124 < r

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   9. Applied rewrites 14.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{{q}^{2}}{\sqrt{{\left(p - r\right)}^{2}}}, 0.5 \cdot \left(\left|r\right| - \left(p + \sqrt{{\left(p - r\right)}^{2}}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

   11. Applied rewrites 44.7%

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

Alternative 2: 48.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|p - r\right|\\ \mathbf{if}\;q \leq 1.75 \cdot 10^{-111}:\\ \;\;\;\;\left(\left|r\right| - r\right) \cdot 0.5\\ \mathbf{elif}\;q \leq 1.66 \cdot 10^{+100}:\\ \;\;\;\;\left(t\_0 - \left(\left|r\right| - p\right)\right) \cdot -0.5 - \frac{q \cdot q}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \mathsf{hypot}\left(r, q + q\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (- p r))))
   (if (<= q 1.75e-111)
     (* (- (fabs r) r) 0.5)
     (if (<= q 1.66e+100)
       (- (* (- t_0 (- (fabs r) p)) -0.5) (/ (* q q) t_0))
       (* 0.5 (- (+ (fabs p) (fabs r)) (hypot r (+ q q))))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs((p - r));
	double tmp;
	if (q <= 1.75e-111) {
		tmp = (fabs(r) - r) * 0.5;
	} else if (q <= 1.66e+100) {
		tmp = ((t_0 - (fabs(r) - p)) * -0.5) - ((q * q) / t_0);
	} else {
		tmp = 0.5 * ((fabs(p) + fabs(r)) - hypot(r, (q + q)));
	}
	return tmp;
}

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double t_0 = Math.abs((p - r));
	double tmp;
	if (q <= 1.75e-111) {
		tmp = (Math.abs(r) - r) * 0.5;
	} else if (q <= 1.66e+100) {
		tmp = ((t_0 - (Math.abs(r) - p)) * -0.5) - ((q * q) / t_0);
	} else {
		tmp = 0.5 * ((Math.abs(p) + Math.abs(r)) - Math.hypot(r, (q + q)));
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	t_0 = math.fabs((p - r))
	tmp = 0
	if q <= 1.75e-111:
		tmp = (math.fabs(r) - r) * 0.5
	elif q <= 1.66e+100:
		tmp = ((t_0 - (math.fabs(r) - p)) * -0.5) - ((q * q) / t_0)
	else:
		tmp = 0.5 * ((math.fabs(p) + math.fabs(r)) - math.hypot(r, (q + q)))
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = abs(Float64(p - r))
	tmp = 0.0
	if (q <= 1.75e-111)
		tmp = Float64(Float64(abs(r) - r) * 0.5);
	elseif (q <= 1.66e+100)
		tmp = Float64(Float64(Float64(t_0 - Float64(abs(r) - p)) * -0.5) - Float64(Float64(q * q) / t_0));
	else
		tmp = Float64(0.5 * Float64(Float64(abs(p) + abs(r)) - hypot(r, Float64(q + q))));
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	t_0 = abs((p - r));
	tmp = 0.0;
	if (q <= 1.75e-111)
		tmp = (abs(r) - r) * 0.5;
	elseif (q <= 1.66e+100)
		tmp = ((t_0 - (abs(r) - p)) * -0.5) - ((q * q) / t_0);
	else
		tmp = 0.5 * ((abs(p) + abs(r)) - hypot(r, (q + q)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[(p - r), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[q, 1.75e-111], N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q, 1.66e+100], N[(N[(N[(t$95$0 - N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] - N[(N[(q * q), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - N[Sqrt[r ^ 2 + N[(q + q), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if q < 1.75e-111

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in p around -inf

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

      1. lower--.f64 N/A

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

      2. lower-fabs.f64 30.4

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

   9. Applied rewrites 30.4%

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

   if 1.75e-111 < q < 1.66e100

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   9. Applied rewrites 14.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{{q}^{2}}{\sqrt{{\left(p - r\right)}^{2}}}, 0.5 \cdot \left(\left|r\right| - \left(p + \sqrt{{\left(p - r\right)}^{2}}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. sub-flip-reverse N/A

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

       5. lower--.f64 14.8

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

   11. Applied rewrites 38.8%

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

   if 1.66e100 < q

   1. Initial program 23.9%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. sub-flip N/A

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

      12. add-flip N/A

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

      13. remove-double-neg N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.3

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

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \mathsf{hypot}\left(r - p, q \cdot 2\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 52.3

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in p around 0

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

   10. Applied rewrites 43.0%

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

Alternative 3: 48.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|p - r\right|\\ \mathbf{if}\;q \leq 1.75 \cdot 10^{-111}:\\ \;\;\;\;\left(\left|r\right| - r\right) \cdot 0.5\\ \mathbf{elif}\;q \leq 1.66 \cdot 10^{+100}:\\ \;\;\;\;\left(t\_0 - \left(\left|r\right| - p\right)\right) \cdot -0.5 - \frac{q \cdot q}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - 2 \cdot q\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (- p r))))
   (if (<= q 1.75e-111)
     (* (- (fabs r) r) 0.5)
     (if (<= q 1.66e+100)
       (- (* (- t_0 (- (fabs r) p)) -0.5) (/ (* q q) t_0))
       (* (/ 1.0 2.0) (- (+ (fabs p) (fabs r)) (* 2.0 q)))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs((p - r));
	double tmp;
	if (q <= 1.75e-111) {
		tmp = (fabs(r) - r) * 0.5;
	} else if (q <= 1.66e+100) {
		tmp = ((t_0 - (fabs(r) - p)) * -0.5) - ((q * q) / t_0);
	} else {
		tmp = (1.0 / 2.0) * ((fabs(p) + fabs(r)) - (2.0 * q));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((p - r))
    if (q <= 1.75d-111) then
        tmp = (abs(r) - r) * 0.5d0
    else if (q <= 1.66d+100) then
        tmp = ((t_0 - (abs(r) - p)) * (-0.5d0)) - ((q * q) / t_0)
    else
        tmp = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) - (2.0d0 * q))
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double t_0 = Math.abs((p - r));
	double tmp;
	if (q <= 1.75e-111) {
		tmp = (Math.abs(r) - r) * 0.5;
	} else if (q <= 1.66e+100) {
		tmp = ((t_0 - (Math.abs(r) - p)) * -0.5) - ((q * q) / t_0);
	} else {
		tmp = (1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) - (2.0 * q));
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	t_0 = math.fabs((p - r))
	tmp = 0
	if q <= 1.75e-111:
		tmp = (math.fabs(r) - r) * 0.5
	elif q <= 1.66e+100:
		tmp = ((t_0 - (math.fabs(r) - p)) * -0.5) - ((q * q) / t_0)
	else:
		tmp = (1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) - (2.0 * q))
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = abs(Float64(p - r))
	tmp = 0.0
	if (q <= 1.75e-111)
		tmp = Float64(Float64(abs(r) - r) * 0.5);
	elseif (q <= 1.66e+100)
		tmp = Float64(Float64(Float64(t_0 - Float64(abs(r) - p)) * -0.5) - Float64(Float64(q * q) / t_0));
	else
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) - Float64(2.0 * q)));
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	t_0 = abs((p - r));
	tmp = 0.0;
	if (q <= 1.75e-111)
		tmp = (abs(r) - r) * 0.5;
	elseif (q <= 1.66e+100)
		tmp = ((t_0 - (abs(r) - p)) * -0.5) - ((q * q) / t_0);
	else
		tmp = (1.0 / 2.0) * ((abs(p) + abs(r)) - (2.0 * q));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[(p - r), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[q, 1.75e-111], N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q, 1.66e+100], N[(N[(N[(t$95$0 - N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] - N[(N[(q * q), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - N[(2.0 * q), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if q < 1.75e-111

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in p around -inf

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

      1. lower--.f64 N/A

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

      2. lower-fabs.f64 30.4

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

   9. Applied rewrites 30.4%

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

   if 1.75e-111 < q < 1.66e100

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   9. Applied rewrites 14.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{{q}^{2}}{\sqrt{{\left(p - r\right)}^{2}}}, 0.5 \cdot \left(\left|r\right| - \left(p + \sqrt{{\left(p - r\right)}^{2}}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. sub-flip-reverse N/A

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

       5. lower--.f64 14.8

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

   11. Applied rewrites 38.8%

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

   if 1.66e100 < q

   1. Initial program 23.9%

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

   2. Taylor expanded in 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 18.1

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

   4. Applied rewrites 18.1%

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

Alternative 4: 48.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 7.9 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| - r, 0.5, \frac{q \cdot q}{p}\right)\\ \mathbf{else}:\\ \;\;\;\;q \cdot \left(0.5 \cdot \frac{\left|r\right| - p}{q} - 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 7.9e+80)
   (fma (- (fabs r) r) 0.5 (/ (* q q) p))
   (* q (- (* 0.5 (/ (- (fabs r) p) q)) 1.0))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 7.9e+80) {
		tmp = fma((fabs(r) - r), 0.5, ((q * q) / p));
	} else {
		tmp = q * ((0.5 * ((fabs(r) - p) / q)) - 1.0);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 7.9e+80)
		tmp = fma(Float64(abs(r) - r), 0.5, Float64(Float64(q * q) / p));
	else
		tmp = Float64(q * Float64(Float64(0.5 * Float64(Float64(abs(r) - p) / q)) - 1.0));
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[q, 7.9e+80], N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] * 0.5 + N[(N[(q * q), $MachinePrecision] / p), $MachinePrecision]), $MachinePrecision], N[(q * N[(N[(0.5 * N[(N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision] / q), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 7.89999999999999999e80

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 38.5

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

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|r\right| - r, \frac{{q}^{2}}{p}\right)} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 38.5

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

       4. lift-pow.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 38.5

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

   11. Applied rewrites 38.5%

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

   if 7.89999999999999999e80 < q

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-fabs.f64 18.0

         \[\leadsto q \cdot \left(0.5 \cdot \frac{\left|r\right| - p}{q} - 1\right) \]

   9. Applied rewrites 18.0%

      \[\leadsto \color{blue}{q \cdot \left(0.5 \cdot \frac{\left|r\right| - p}{q} - 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 46.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 7.9 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| - r, 0.5, \frac{q \cdot q}{p}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - 2 \cdot q\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 7.9e+80)
   (fma (- (fabs r) r) 0.5 (/ (* q q) p))
   (* (/ 1.0 2.0) (- (+ (fabs p) (fabs r)) (* 2.0 q)))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 7.9e+80) {
		tmp = fma((fabs(r) - r), 0.5, ((q * q) / p));
	} else {
		tmp = (1.0 / 2.0) * ((fabs(p) + fabs(r)) - (2.0 * q));
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 7.9e+80)
		tmp = fma(Float64(abs(r) - r), 0.5, Float64(Float64(q * q) / p));
	else
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) - Float64(2.0 * q)));
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[q, 7.9e+80], N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] * 0.5 + N[(N[(q * q), $MachinePrecision] / p), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - N[(2.0 * q), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 7.89999999999999999e80

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 38.5

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

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|r\right| - r, \frac{{q}^{2}}{p}\right)} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 38.5

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

       4. lift-pow.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 38.5

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

   11. Applied rewrites 38.5%

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

   if 7.89999999999999999e80 < q

   1. Initial program 23.9%

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

   2. Taylor expanded in 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 18.1

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

   4. Applied rewrites 18.1%

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

Alternative 6: 45.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 2.35 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| - r, 0.5, \frac{q \cdot q}{p}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 2.35e+69) (fma (- (fabs r) r) 0.5 (/ (* q q) p)) (- q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 2.35e+69) {
		tmp = fma((fabs(r) - r), 0.5, ((q * q) / p));
	} else {
		tmp = -q;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 2.35e+69)
		tmp = fma(Float64(abs(r) - r), 0.5, Float64(Float64(q * q) / p));
	else
		tmp = Float64(-q);
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[q, 2.35e+69], N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] * 0.5 + N[(N[(q * q), $MachinePrecision] / p), $MachinePrecision]), $MachinePrecision], (-q)]

Derivation

1. Split input into 2 regimes

2. if q < 2.34999999999999998e69

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 38.5

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

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|r\right| - r, \frac{{q}^{2}}{p}\right)} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 38.5

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

       4. lift-pow.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 38.5

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

   11. Applied rewrites 38.5%

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

   if 2.34999999999999998e69 < q

   1. Initial program 23.9%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.1

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

   4. Applied rewrites 19.1%

      \[\leadsto \color{blue}{-1 \cdot q} \]
   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. *-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(q \cdot 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(q \cdot \left(\mathsf{neg}\left(-1\right)\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(q \cdot -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot q\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot q\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(-1 \cdot q\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(-1 \cdot q\right)\right) \]

      10. *-commutative N/A

          \[\leadsto -\left(\mathsf{neg}\left(q \cdot -1\right)\right) \]

      11. distribute-rgt-neg-out N/A

          \[\leadsto -q \cdot \left(\mathsf{neg}\left(-1\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto -q \cdot 1 \]

      13. *-rgt-identity 19.1

          \[\leadsto -q \]

   6. Applied rewrites 19.1%

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

Alternative 7: 43.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(\left|r\right| - r\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 2e-11) (* (- (fabs r) r) 0.5) (- q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 2e-11) {
		tmp = (fabs(r) - r) * 0.5;
	} else {
		tmp = -q;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (q <= 2d-11) then
        tmp = (abs(r) - r) * 0.5d0
    else
        tmp = -q
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (q <= 2e-11) {
		tmp = (Math.abs(r) - r) * 0.5;
	} else {
		tmp = -q;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if q <= 2e-11:
		tmp = (math.fabs(r) - r) * 0.5
	else:
		tmp = -q
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 2e-11)
		tmp = Float64(Float64(abs(r) - r) * 0.5);
	else
		tmp = Float64(-q);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (q <= 2e-11)
		tmp = (abs(r) - r) * 0.5;
	else
		tmp = -q;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[q, 2e-11], N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision], (-q)]

Derivation

1. Split input into 2 regimes

2. if q < 1.99999999999999988e-11

   1. Initial program 23.9%

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

   2. Taylor expanded in r around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.3

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

   6. Applied rewrites 2.3%

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

   7. Taylor expanded in p around -inf

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

      1. lower--.f64 N/A

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

      2. lower-fabs.f64 30.4

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

   9. Applied rewrites 30.4%

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

   if 1.99999999999999988e-11 < q

   1. Initial program 23.9%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.1

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

   4. Applied rewrites 19.1%

      \[\leadsto \color{blue}{-1 \cdot q} \]
   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. *-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(q \cdot 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(q \cdot \left(\mathsf{neg}\left(-1\right)\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(q \cdot -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot q\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot q\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(-1 \cdot q\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(-1 \cdot q\right)\right) \]

      10. *-commutative N/A

          \[\leadsto -\left(\mathsf{neg}\left(q \cdot -1\right)\right) \]

      11. distribute-rgt-neg-out N/A

          \[\leadsto -q \cdot \left(\mathsf{neg}\left(-1\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto -q \cdot 1 \]

      13. *-rgt-identity 19.1

          \[\leadsto -q \]

   6. Applied rewrites 19.1%

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

Alternative 8: 19.1% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

2. Taylor expanded in q around inf

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

   1. lower-*.f64 19.1

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

4. Applied rewrites 19.1%

   \[\leadsto \color{blue}{-1 \cdot q} \]
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. *-rgt-identity N/A

      \[\leadsto \mathsf{neg}\left(q \cdot 1\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(q \cdot \left(\mathsf{neg}\left(-1\right)\right)\right) \]

   5. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(q \cdot -1\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot q\right)\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot q\right)\right)\right) \]

   8. lower-neg.f64 N/A

      \[\leadsto -\left(\mathsf{neg}\left(-1 \cdot q\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto -\left(\mathsf{neg}\left(-1 \cdot q\right)\right) \]

   10. *-commutative N/A

       \[\leadsto -\left(\mathsf{neg}\left(q \cdot -1\right)\right) \]

   11. distribute-rgt-neg-out N/A

       \[\leadsto -q \cdot \left(\mathsf{neg}\left(-1\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto -q \cdot 1 \]

   13. *-rgt-identity 19.1

       \[\leadsto -q \]

6. Applied rewrites 19.1%

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

Reproduce

herbie shell --seed 2025152 
(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)))))))