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

Percentage Accurate: 45.7% → 79.7%

Time: 7.5s

Alternatives: 11

Speedup: 12.5×

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: 45.7% 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: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right)\\ \mathbf{if}\;t\_0 \leq 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| + r\right) + \left|r\right|, 0.5, -0.5 \cdot p\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0
         (*
          (/ 1.0 2.0)
          (+
           (+ (fabs p) (fabs r))
           (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0))))))))
   (if (<= t_0 1e+143) t_0 (fma (+ (+ (fabs p) r) (fabs r)) 0.5 (* -0.5 p)))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = (1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
	double tmp;
	if (t_0 <= 1e+143) {
		tmp = t_0;
	} else {
		tmp = fma(((fabs(p) + r) + fabs(r)), 0.5, (-0.5 * p));
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = 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))))))
	tmp = 0.0
	if (t_0 <= 1e+143)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(abs(p) + r) + abs(r)), 0.5, Float64(-0.5 * 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_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 1e+143], t$95$0, N[(N[(N[(N[Abs[p], $MachinePrecision] + r), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-0.5 * p), $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))))))) < 1e143

   1. Initial program 96.2%

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

   if 1e143 < (*.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 10.0%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 28.9

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 35.4%

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

   10. Applied rewrites 17.8%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 35.8%

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

Alternative 2: 74.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 1.75 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| + r\right) + \left|r\right|, 0.5, -0.5 \cdot p\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 1.75e+56)
   (fma (+ (+ (fabs p) r) (fabs r)) 0.5 (* -0.5 p))
   (fma 0.5 (+ (fabs r) (fabs p)) q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 1.75e+56) {
		tmp = fma(((fabs(p) + r) + fabs(r)), 0.5, (-0.5 * p));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 1.75e+56)
		tmp = fma(Float64(Float64(abs(p) + r) + abs(r)), 0.5, Float64(-0.5 * p));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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, 1.75e+56], N[(N[(N[(N[Abs[p], $MachinePrecision] + r), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-0.5 * p), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.75e56

   1. Initial program 50.3%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 39.7%

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

   10. Applied rewrites 19.6%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 39.9%

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

   if 1.75e56 < q

   1. Initial program 33.6%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 70.8%

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

Alternative 3: 56.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -9.2 \cdot 10^{-257}:\\ \;\;\;\;-0.5 \cdot \left(\left(p - \left|r\right|\right) - \left|p\right|\right)\\ \mathbf{elif}\;r \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\right)\\ \mathbf{else}:\\ \;\;\;\;\left(p - 2 \cdot r\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r -9.2e-257)
   (* -0.5 (- (- p (fabs r)) (fabs p)))
   (if (<= r 9.5e+32)
     (fma 0.5 (+ (fabs r) (fabs p)) q)
     (* (- p (* 2.0 r)) -0.5))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= -9.2e-257) {
		tmp = -0.5 * ((p - fabs(r)) - fabs(p));
	} else if (r <= 9.5e+32) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q);
	} else {
		tmp = (p - (2.0 * r)) * -0.5;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= -9.2e-257)
		tmp = Float64(-0.5 * Float64(Float64(p - abs(r)) - abs(p)));
	elseif (r <= 9.5e+32)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q);
	else
		tmp = Float64(Float64(p - Float64(2.0 * 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, -9.2e-257], N[(-0.5 * N[(N[(p - N[Abs[r], $MachinePrecision]), $MachinePrecision] - N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 9.5e+32], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q), $MachinePrecision], N[(N[(p - N[(2.0 * r), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -9.2000000000000001e-257

   1. Initial program 40.0%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 21.1

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 21.1%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 26.4%

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

   if -9.2000000000000001e-257 < r < 9.50000000000000006e32

   1. Initial program 61.1%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 32.2%

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

   if 9.50000000000000006e32 < r

   1. Initial program 34.8%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.4%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 87.3%

       \[\leadsto \left(p - 2 \cdot r\right) \cdot -0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 1.75 \cdot 10^{+56}:\\ \;\;\;\;-0.5 \cdot \left(p - \left(\left(r + \left|r\right|\right) + \left|p\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 1.75e+56)
   (* -0.5 (- p (+ (+ r (fabs r)) (fabs p))))
   (fma 0.5 (+ (fabs r) (fabs p)) q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 1.75e+56) {
		tmp = -0.5 * (p - ((r + fabs(r)) + fabs(p)));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 1.75e+56)
		tmp = Float64(-0.5 * Float64(p - Float64(Float64(r + abs(r)) + abs(p))));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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, 1.75e+56], N[(-0.5 * N[(p - N[(N[(r + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.75e56

   1. Initial program 50.3%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 39.7%

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

   if 1.75e56 < q

   1. Initial program 33.6%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 70.8%

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

Alternative 5: 47.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\right)\\ \mathbf{else}:\\ \;\;\;\;\left(p - 2 \cdot r\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r 9.5e+32)
   (fma 0.5 (+ (fabs r) (fabs p)) q)
   (* (- p (* 2.0 r)) -0.5)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 9.5e+32) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q);
	} else {
		tmp = (p - (2.0 * r)) * -0.5;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 9.5e+32)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q);
	else
		tmp = Float64(Float64(p - Float64(2.0 * 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, 9.5e+32], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q), $MachinePrecision], N[(N[(p - N[(2.0 * r), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.50000000000000006e32

   1. Initial program 49.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 30.4%

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

   if 9.50000000000000006e32 < r

   1. Initial program 34.8%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.4%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 87.3%

       \[\leadsto \left(p - 2 \cdot r\right) \cdot -0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 42.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, q\right)\\ \mathbf{else}:\\ \;\;\;\;\left(p - 2 \cdot r\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r 9.5e+32) (fma 0.5 r q) (* (- p (* 2.0 r)) -0.5)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 9.5e+32) {
		tmp = fma(0.5, r, q);
	} else {
		tmp = (p - (2.0 * r)) * -0.5;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 9.5e+32)
		tmp = fma(0.5, r, q);
	else
		tmp = Float64(Float64(p - Float64(2.0 * 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, 9.5e+32], N[(0.5 * r + q), $MachinePrecision], N[(N[(p - N[(2.0 * r), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.50000000000000006e32

   1. Initial program 49.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 30.4%

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

   10. Applied rewrites 23.2%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 20.6%

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

   if 9.50000000000000006e32 < r

   1. Initial program 34.8%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.4%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 87.3%

       \[\leadsto \left(p - 2 \cdot r\right) \cdot -0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 42.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, q\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot r\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r 9.5e+32) (fma 0.5 r q) (* (* -2.0 r) -0.5)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 9.5e+32) {
		tmp = fma(0.5, r, q);
	} else {
		tmp = (-2.0 * r) * -0.5;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 9.5e+32)
		tmp = fma(0.5, r, q);
	else
		tmp = Float64(Float64(-2.0 * 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, 9.5e+32], N[(0.5 * r + q), $MachinePrecision], N[(N[(-2.0 * r), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.50000000000000006e32

   1. Initial program 49.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 30.4%

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

   10. Applied rewrites 23.2%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 20.6%

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

   if 9.50000000000000006e32 < r

   1. Initial program 34.8%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.4%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 87.6%

       \[\leadsto \left(-2 \cdot r\right) \cdot -0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 13.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.65 \cdot 10^{-38}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (if (<= r 1.65e-38) (* -0.5 p) (* 0.5 r)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 1.65e-38) {
		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 <= 1.65d-38) 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 <= 1.65e-38) {
		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 <= 1.65e-38:
		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 <= 1.65e-38)
		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 <= 1.65e-38)
		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, 1.65e-38], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.6500000000000001e-38

   1. Initial program 47.3%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 6.1

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

   5. Applied rewrites 6.1%

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

   if 1.6500000000000001e-38 < r

   1. Initial program 46.2%

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

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 14.6

         \[\leadsto \color{blue}{0.5 \cdot r} \]

   5. Applied rewrites 14.6%

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

Alternative 9: 23.5% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.1%

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

3. Taylor expanded in q around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-fabs.f64 26.7

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

5. Applied rewrites 26.7%

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

6. Taylor expanded in p around 0

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

8. Applied rewrites 28.8%

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

10. Applied rewrites 22.6%

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

11. Taylor expanded in p around 0

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

13. Applied rewrites 20.4%

    \[\leadsto \mathsf{fma}\left(0.5, r, q\right) \]
14. Add Preprocessing

Alternative 10: 8.7% accurate, 41.7× 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 47.1%

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

3. Taylor expanded in p around -inf

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

   1. lower-*.f64 5.3

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

5. Applied rewrites 5.3%

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

Alternative 11: 18.9% accurate, 83.3× 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 47.1%

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

3. Taylor expanded in q around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 18.7

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

5. Applied rewrites 18.7%

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

Reproduce

herbie shell --seed 2025006 
(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)))))))