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

Percentage Accurate: 45.3% → 80.0%

Time: 3.6s

Alternatives: 11

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 45.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \leq 10^{+154}:\\ \;\;\;\;\left(\left(\left|r\right| + \sqrt{\mathsf{fma}\left(4 \cdot q, q, \left(r - p\right) \cdot \left(r - p\right)\right)}\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<=
      (*
       (/ 1.0 2.0)
       (+
        (+ (fabs p) (fabs r))
        (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0))))))
      1e+154)
   (*
    (+ (+ (fabs r) (sqrt (fma (* 4.0 q) q (* (- r p) (- r p))))) (fabs p))
    0.5)
   (fma 0.5 r (* 0.5 (+ (fabs p) (+ (fabs r) (* -1.0 p)))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (((1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))))) <= 1e+154) {
		tmp = ((fabs(r) + sqrt(fma((4.0 * q), q, ((r - p) * (r - p))))) + fabs(p)) * 0.5;
	} else {
		tmp = fma(0.5, r, (0.5 * (fabs(p) + (fabs(r) + (-1.0 * p)))));
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (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)))))) <= 1e+154)
		tmp = Float64(Float64(Float64(abs(r) + sqrt(fma(Float64(4.0 * q), q, Float64(Float64(r - p) * Float64(r - p))))) + abs(p)) * 0.5);
	else
		tmp = fma(0.5, r, Float64(0.5 * Float64(abs(p) + Float64(abs(r) + Float64(-1.0 * p)))));
	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[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], 1e+154], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Sqrt[N[(N[(4.0 * q), $MachinePrecision] * q + N[(N[(r - p), $MachinePrecision] * N[(r - p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * r + N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] + N[(-1.0 * p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))))))) < 1.00000000000000004e154

   1. Initial program 45.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.3

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

   3. Applied rewrites 45.3%

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

   if 1.00000000000000004e154 < (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))))))

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

Alternative 2: 74.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 3.15 \cdot 10^{+47}:\\ \;\;\;\;\left(r - \left(\left(p - \left|p\right|\right) - \left|r\right|\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, q\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 3.15e+47)
   (* (- r (- (- p (fabs p)) (fabs r))) 0.5)
   (fma (+ (fabs r) (fabs p)) 0.5 q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 3.15e+47) {
		tmp = (r - ((p - fabs(p)) - fabs(r))) * 0.5;
	} else {
		tmp = fma((fabs(r) + fabs(p)), 0.5, q);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 3.15e+47)
		tmp = Float64(Float64(r - Float64(Float64(p - abs(p)) - abs(r))) * 0.5);
	else
		tmp = fma(Float64(abs(r) + abs(p)), 0.5, 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, 3.15e+47], N[(N[(r - N[(N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + q), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3.15000000000000002e47

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 67.7%

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

   if 3.15000000000000002e47 < q

   1. Initial program 45.3%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.0

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

   4. Applied rewrites 26.0%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 28.5%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 28.5

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

   8. Applied rewrites 28.5%

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

Alternative 3: 54.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\left(\left|r\right| - \left(p - \left|p\right|\right)\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 6.2 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, q\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r -3.7e-233)
   (* (- (fabs r) (- p (fabs p))) 0.5)
   (if (<= r 6.2e+70)
     (fma (+ (fabs r) (fabs p)) 0.5 q)
     (fma 0.5 r (* 0.5 (+ (fabs p) (fabs r)))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= -3.7e-233) {
		tmp = (fabs(r) - (p - fabs(p))) * 0.5;
	} else if (r <= 6.2e+70) {
		tmp = fma((fabs(r) + fabs(p)), 0.5, q);
	} else {
		tmp = fma(0.5, r, (0.5 * (fabs(p) + fabs(r))));
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= -3.7e-233)
		tmp = Float64(Float64(abs(r) - Float64(p - abs(p))) * 0.5);
	elseif (r <= 6.2e+70)
		tmp = fma(Float64(abs(r) + abs(p)), 0.5, q);
	else
		tmp = fma(0.5, r, Float64(0.5 * Float64(abs(p) + abs(r))));
	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[r, -3.7e-233], N[(N[(N[Abs[r], $MachinePrecision] - N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 6.2e+70], N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + q), $MachinePrecision], N[(0.5 * r + N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -3.6999999999999998e-233

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right) \]

   7. Applied rewrites 40.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip-reverse N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 40.5

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

   9. Applied rewrites 40.5%

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

   if -3.6999999999999998e-233 < r < 6.2000000000000006e70

   1. Initial program 45.3%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.0

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

   4. Applied rewrites 26.0%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 28.5%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 28.5

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

   8. Applied rewrites 28.5%

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

   if 6.2000000000000006e70 < r

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in p around 0

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

      1. lower-fabs.f64 40.6

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

   10. Applied rewrites 40.6%

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

Alternative 4: 54.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\left(\left|r\right| - \left(p - \left|p\right|\right)\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 6.2 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, q\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r + \left(\left|p\right| + \left|r\right|\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r -3.7e-233)
   (* (- (fabs r) (- p (fabs p))) 0.5)
   (if (<= r 6.2e+70)
     (fma (+ (fabs r) (fabs p)) 0.5 q)
     (* (+ r (+ (fabs p) (fabs r))) 0.5))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= -3.7e-233) {
		tmp = (fabs(r) - (p - fabs(p))) * 0.5;
	} else if (r <= 6.2e+70) {
		tmp = fma((fabs(r) + fabs(p)), 0.5, q);
	} else {
		tmp = (r + (fabs(p) + fabs(r))) * 0.5;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= -3.7e-233)
		tmp = Float64(Float64(abs(r) - Float64(p - abs(p))) * 0.5);
	elseif (r <= 6.2e+70)
		tmp = fma(Float64(abs(r) + abs(p)), 0.5, q);
	else
		tmp = Float64(Float64(r + Float64(abs(p) + abs(r))) * 0.5);
	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[r, -3.7e-233], N[(N[(N[Abs[r], $MachinePrecision] - N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 6.2e+70], N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + q), $MachinePrecision], N[(N[(r + N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -3.6999999999999998e-233

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right) \]

   7. Applied rewrites 40.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip-reverse N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 40.5

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

   9. Applied rewrites 40.5%

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

   if -3.6999999999999998e-233 < r < 6.2000000000000006e70

   1. Initial program 45.3%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.0

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

   4. Applied rewrites 26.0%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 28.5%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 28.5

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

   8. Applied rewrites 28.5%

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

   if 6.2000000000000006e70 < r

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 67.7%

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

   10. Taylor expanded in p around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-fabs.f64 N/A

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

       4. lower-fabs.f64 40.5

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

   12. Applied rewrites 40.5%

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

Alternative 5: 45.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -5.6 \cdot 10^{+89}:\\ \;\;\;\;\left(\left|r\right| - \left(p - \left|p\right|\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, q\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= p -5.6e+89)
   (* (- (fabs r) (- p (fabs p))) 0.5)
   (fma (+ (fabs r) (fabs p)) 0.5 q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (p <= -5.6e+89) {
		tmp = (fabs(r) - (p - fabs(p))) * 0.5;
	} else {
		tmp = fma((fabs(r) + fabs(p)), 0.5, q);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (p <= -5.6e+89)
		tmp = Float64(Float64(abs(r) - Float64(p - abs(p))) * 0.5);
	else
		tmp = fma(Float64(abs(r) + abs(p)), 0.5, 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[p, -5.6e+89], N[(N[(N[Abs[r], $MachinePrecision] - N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + q), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -5.5999999999999996e89

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right) \]

   7. Applied rewrites 40.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-flip-reverse N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 40.5

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

   9. Applied rewrites 40.5%

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

   if -5.5999999999999996e89 < p

   1. Initial program 45.3%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.0

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

   4. Applied rewrites 26.0%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 28.5%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 28.5

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

   8. Applied rewrites 28.5%

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

Alternative 6: 28.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, q\right) \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (fma (+ (fabs r) (fabs p)) 0.5 q))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	return fma((fabs(r) + fabs(p)), 0.5, q);
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	return fma(Float64(abs(r) + abs(p)), 0.5, q)
end

Wolfram

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

Derivation

1. Initial program 45.3%

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

2. Taylor expanded in q around inf

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

   1. metadata-eval N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. lower-fabs.f64 26.0

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

4. Applied rewrites 26.0%

   \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
5. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. sum-to-mult-rev N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

6. Applied rewrites 28.5%

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

   1. metadata-eval N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. metadata-eval 28.5

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

8. Applied rewrites 28.5%

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

Alternative 7: 26.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 3.4 \cdot 10^{-65}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\\ \mathbf{else}:\\ \;\;\;\;q \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 3.4e-65) (* 0.5 (+ (fabs p) (fabs r))) (* q 1.0)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 3.4e-65) {
		tmp = 0.5 * (fabs(p) + fabs(r));
	} else {
		tmp = q * 1.0;
	}
	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 <= 3.4d-65) then
        tmp = 0.5d0 * (abs(p) + abs(r))
    else
        tmp = q * 1.0d0
    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 <= 3.4e-65) {
		tmp = 0.5 * (Math.abs(p) + Math.abs(r));
	} else {
		tmp = q * 1.0;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if q <= 3.4e-65:
		tmp = 0.5 * (math.fabs(p) + math.fabs(r))
	else:
		tmp = q * 1.0
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 3.4e-65)
		tmp = Float64(0.5 * Float64(abs(p) + abs(r)));
	else
		tmp = Float64(q * 1.0);
	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 <= 3.4e-65)
		tmp = 0.5 * (abs(p) + abs(r));
	else
		tmp = q * 1.0;
	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, 3.4e-65], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(q * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3.39999999999999987e-65

   1. Initial program 45.3%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.0

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

   4. Applied rewrites 26.0%

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

   5. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-fabs.f64 14.4

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

   7. Applied rewrites 14.4%

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

   if 3.39999999999999987e-65 < q

   1. Initial program 45.3%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.0

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

   4. Applied rewrites 26.0%

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

   5. Taylor expanded in q around inf

      \[\leadsto q \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 17.8%

      \[\leadsto q \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 20.0% accurate, 7.3× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.7 \cdot 10^{+167}:\\ \;\;\;\;q \cdot 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (if (<= r 1.7e+167) (* q 1.0) (* 0.5 r)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 1.7e+167) {
		tmp = q * 1.0;
	} else {
		tmp = 0.5 * r;
	}
	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 (r <= 1.7d+167) then
        tmp = q * 1.0d0
    else
        tmp = 0.5d0 * r
    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 (r <= 1.7e+167) {
		tmp = q * 1.0;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 1.7e+167:
		tmp = q * 1.0
	else:
		tmp = 0.5 * r
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 1.7e+167)
		tmp = Float64(q * 1.0);
	else
		tmp = Float64(0.5 * r);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 1.7e+167)
		tmp = q * 1.0;
	else
		tmp = 0.5 * r;
	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[r, 1.7e+167], N[(q * 1.0), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.7e167

   1. Initial program 45.3%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.0

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

   4. Applied rewrites 26.0%

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

   5. Taylor expanded in q around inf

      \[\leadsto q \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 17.8%

      \[\leadsto q \cdot 1 \]

   if 1.7e167 < r

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot r \]

      2. lower-*.f64 N/A

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

      3. metadata-eval 8.6

         \[\leadsto 0.5 \cdot r \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 19.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 10^{-111}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (if (<= r 1e-111) (* -0.5 p) (* 0.5 r)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 1e-111) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	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 (r <= 1d-111) then
        tmp = (-0.5d0) * p
    else
        tmp = 0.5d0 * r
    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 (r <= 1e-111) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 1e-111:
		tmp = -0.5 * p
	else:
		tmp = 0.5 * r
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 1e-111)
		tmp = Float64(-0.5 * p);
	else
		tmp = Float64(0.5 * r);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 1e-111)
		tmp = -0.5 * p;
	else
		tmp = 0.5 * r;
	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[r, 1e-111], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.00000000000000009e-111

   1. Initial program 45.3%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 8.6

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

   4. Applied rewrites 8.6%

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

   if 1.00000000000000009e-111 < r

   1. Initial program 45.3%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot r \]

      2. lower-*.f64 N/A

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

      3. metadata-eval 8.6

         \[\leadsto 0.5 \cdot r \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 13.0% accurate, 14.4× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ -0.5 \cdot p \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (* -0.5 p))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	return -0.5 * p;
}

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 = (-0.5d0) * p
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	return -0.5 * p;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	return -0.5 * p

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	return Float64(-0.5 * p)
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp = code(p, r, q)
	tmp = -0.5 * p;
end

Wolfram

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

Derivation

1. Initial program 45.3%

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

2. Taylor expanded in p around -inf

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

   1. lower-*.f64 8.6

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

4. Applied rewrites 8.6%

   \[\leadsto \color{blue}{-0.5 \cdot p} \]
5. Add Preprocessing

Alternative 11: 8.6% 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 45.3%

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

2. Taylor expanded in 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. lower-neg.f64 19.1

      \[\leadsto -q \]

6. Applied rewrites 19.1%

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

Reproduce

herbie shell --seed 2025149 
(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)))))))