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

Percentage Accurate: 45.8% → 73.7%

Time: 2.9s

Alternatives: 4

Speedup: N/A×

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: 45.8% 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: 73.7% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ t_1 := \frac{1}{2} \cdot \left(t\_0 + \left(\mathsf{fma}\left(\frac{r}{p}, -1, 1\right) \cdot p\right) \cdot -1\right)\\ \mathbf{if}\;q\_m \leq 3.1 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;q\_m \leq 1.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q\_m}^{2}}\right)\\ \mathbf{elif}\;q\_m \leq 1.9 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + q\_m \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (+ (fabs p) (fabs r)))
        (t_1 (* (/ 1.0 2.0) (+ t_0 (* (* (fma (/ r p) -1.0 1.0) p) -1.0)))))
   (if (<= q_m 3.1e-14)
     t_1
     (if (<= q_m 1.7e+68)
       (*
        (/ 1.0 2.0)
        (+ t_0 (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q_m 2.0))))))
       (if (<= q_m 1.9e+108) t_1 (* (/ 1.0 2.0) (+ t_0 (* q_m 2.0))))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(p) + fabs(r);
	double t_1 = (1.0 / 2.0) * (t_0 + ((fma((r / p), -1.0, 1.0) * p) * -1.0));
	double tmp;
	if (q_m <= 3.1e-14) {
		tmp = t_1;
	} else if (q_m <= 1.7e+68) {
		tmp = (1.0 / 2.0) * (t_0 + sqrt((pow((p - r), 2.0) + (4.0 * pow(q_m, 2.0)))));
	} else if (q_m <= 1.9e+108) {
		tmp = t_1;
	} else {
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0));
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(abs(p) + abs(r))
	t_1 = Float64(Float64(1.0 / 2.0) * Float64(t_0 + Float64(Float64(fma(Float64(r / p), -1.0, 1.0) * p) * -1.0)))
	tmp = 0.0
	if (q_m <= 3.1e-14)
		tmp = t_1;
	elseif (q_m <= 1.7e+68)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q_m ^ 2.0))))));
	elseif (q_m <= 1.9e+108)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + Float64(q_m * 2.0)));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[(N[(N[(N[(r / p), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] * p), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 3.1e-14], t$95$1, If[LessEqual[q$95$m, 1.7e+68], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 1.9e+108], t$95$1, N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 3.10000000000000004e-14 or 1.70000000000000008e68 < q < 1.90000000000000004e108

   1. Initial program 46.3%

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

   3. Taylor expanded in p around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 31.4

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

   5. Applied rewrites 31.4%

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

   if 3.10000000000000004e-14 < q < 1.70000000000000008e68

   1. Initial program 69.1%

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

   if 1.90000000000000004e108 < q

   1. Initial program 19.5%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.9

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

   5. Applied rewrites 76.9%

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

Alternative 2: 71.6% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;q\_m \leq 3.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + \left(\mathsf{fma}\left(\frac{r}{p}, -1, 1\right) \cdot p\right) \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + q\_m \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (+ (fabs p) (fabs r))))
   (if (<= q_m 3.8e-23)
     (* (/ 1.0 2.0) (+ t_0 (* (* (fma (/ r p) -1.0 1.0) p) -1.0)))
     (* (/ 1.0 2.0) (+ t_0 (* q_m 2.0))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(p) + fabs(r);
	double tmp;
	if (q_m <= 3.8e-23) {
		tmp = (1.0 / 2.0) * (t_0 + ((fma((r / p), -1.0, 1.0) * p) * -1.0));
	} else {
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0));
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (q_m <= 3.8e-23)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + Float64(Float64(fma(Float64(r / p), -1.0, 1.0) * p) * -1.0)));
	else
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + Float64(q_m * 2.0)));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 3.8e-23], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[(N[(N[(N[(r / p), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] * p), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 3.80000000000000011e-23

   1. Initial program 47.2%

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

   3. Taylor expanded in p around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 30.8

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

   5. Applied rewrites 30.8%

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

   if 3.80000000000000011e-23 < q

   1. Initial program 29.4%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

Alternative 3: 60.3% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 3.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q\_m \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\mathsf{fma}\left(\frac{p}{r}, -1, \frac{r + p}{r}\right) + 1\right) \cdot r\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 3.5e+31)
   (* (/ 1.0 2.0) (+ (+ (fabs p) (fabs r)) (* q_m 2.0)))
   (* (/ 1.0 2.0) (* (+ (fma (/ p r) -1.0 (/ (+ r p) r)) 1.0) r))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 3.5e+31) {
		tmp = (1.0 / 2.0) * ((fabs(p) + fabs(r)) + (q_m * 2.0));
	} else {
		tmp = (1.0 / 2.0) * ((fma((p / r), -1.0, ((r + p) / r)) + 1.0) * r);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 3.5e+31)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) + Float64(q_m * 2.0)));
	else
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(fma(Float64(p / r), -1.0, Float64(Float64(r + p) / r)) + 1.0) * r));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 3.5e+31], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(N[(p / r), $MachinePrecision] * -1.0 + N[(N[(r + p), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.5e31

   1. Initial program 44.9%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.6

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

   5. Applied rewrites 31.6%

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

   if 3.5e31 < r

   1. Initial program 33.7%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.6%

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

Alternative 4: 35.1% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \frac{1}{2} \cdot \left(\left(\mathsf{fma}\left(\frac{p}{r}, -1, \frac{r + p}{r}\right) + 1\right) \cdot r\right) \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (* (/ 1.0 2.0) (* (+ (fma (/ p r) -1.0 (/ (+ r p) r)) 1.0) r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return (1.0 / 2.0) * ((fma((p / r), -1.0, ((r + p) / r)) + 1.0) * r);
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(Float64(1.0 / 2.0) * Float64(Float64(fma(Float64(p / r), -1.0, Float64(Float64(r + p) / r)) + 1.0) * r))
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(N[(p / r), $MachinePrecision] * -1.0 + N[(N[(r + p), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.5%

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

3. Taylor expanded in r around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 17.4%

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

Reproduce

herbie shell --seed 2025064 
(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)))))))