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

Percentage Accurate: 24.5% → 57.4%

Time: 8.2s

Alternatives: 5

Speedup: 83.3×

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

real(8) function code(p, r, q)
    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: 24.5% 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

real(8) function code(p, r, q)
    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: 57.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.5 \cdot 10^{-229}:\\ \;\;\;\;\left(-p\right) \cdot \left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p} \cdot -0.5 - 0.5\right)\\ \mathbf{elif}\;q\_m \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{\left|r\right| + \left(\left|p\right| + p\right)}{r} \cdot 0.5 - 0.5\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= q_m 1.5e-229)
   (* (- p) (- (* (/ (+ (- (fabs r) r) (fabs p)) p) -0.5) 0.5))
   (if (<= q_m 8.5e-14)
     (* (- (* (/ (+ (fabs r) (+ (fabs p) p)) r) 0.5) 0.5) r)
     (- q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.5e-229) {
		tmp = -p * (((((fabs(r) - r) + fabs(p)) / p) * -0.5) - 0.5);
	} else if (q_m <= 8.5e-14) {
		tmp = ((((fabs(r) + (fabs(p) + p)) / r) * 0.5) - 0.5) * r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 1.5d-229) then
        tmp = -p * (((((abs(r) - r) + abs(p)) / p) * (-0.5d0)) - 0.5d0)
    else if (q_m <= 8.5d-14) then
        tmp = ((((abs(r) + (abs(p) + p)) / r) * 0.5d0) - 0.5d0) * r
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.5e-229) {
		tmp = -p * (((((Math.abs(r) - r) + Math.abs(p)) / p) * -0.5) - 0.5);
	} else if (q_m <= 8.5e-14) {
		tmp = ((((Math.abs(r) + (Math.abs(p) + p)) / r) * 0.5) - 0.5) * r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 1.5e-229:
		tmp = -p * (((((math.fabs(r) - r) + math.fabs(p)) / p) * -0.5) - 0.5)
	elif q_m <= 8.5e-14:
		tmp = ((((math.fabs(r) + (math.fabs(p) + p)) / r) * 0.5) - 0.5) * r
	else:
		tmp = -q_m
	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 (q_m <= 1.5e-229)
		tmp = Float64(Float64(-p) * Float64(Float64(Float64(Float64(Float64(abs(r) - r) + abs(p)) / p) * -0.5) - 0.5));
	elseif (q_m <= 8.5e-14)
		tmp = Float64(Float64(Float64(Float64(Float64(abs(r) + Float64(abs(p) + p)) / r) * 0.5) - 0.5) * r);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 1.5e-229)
		tmp = -p * (((((abs(r) - r) + abs(p)) / p) * -0.5) - 0.5);
	elseif (q_m <= 8.5e-14)
		tmp = ((((abs(r) + (abs(p) + p)) / r) * 0.5) - 0.5) * r;
	else
		tmp = -q_m;
	end
	tmp_2 = 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[q$95$m, 1.5e-229], N[((-p) * N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 8.5e-14], N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * r), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if q < 1.50000000000000001e-229

   1. Initial program 19.4%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate--l+ N/A

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

      4. div-add N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. div-add N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 19.1%

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

   if 1.50000000000000001e-229 < q < 8.50000000000000038e-14

   1. Initial program 20.4%

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

   3. Taylor expanded in r around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 14.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) \cdot 0.5} \]

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 16.1%

      \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]

   9. Taylor expanded in r around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 19.1%

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

   if 8.50000000000000038e-14 < q

   1. Initial program 20.7%

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

   3. Taylor expanded in 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 66.8

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

   5. Applied rewrites 66.8%

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

Alternative 2: 47.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 8.2 \cdot 10^{-112}:\\ \;\;\;\;{2}^{-1} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \left(r - p\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= q_m 8.2e-112)
   (* (pow 2.0 -1.0) (- (+ (fabs p) (fabs r)) (- r p)))
   (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 8.2e-112) {
		tmp = pow(2.0, -1.0) * ((fabs(p) + fabs(r)) - (r - p));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 8.2d-112) then
        tmp = (2.0d0 ** (-1.0d0)) * ((abs(p) + abs(r)) - (r - p))
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 8.2e-112) {
		tmp = Math.pow(2.0, -1.0) * ((Math.abs(p) + Math.abs(r)) - (r - p));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 8.2e-112:
		tmp = math.pow(2.0, -1.0) * ((math.fabs(p) + math.fabs(r)) - (r - p))
	else:
		tmp = -q_m
	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 (q_m <= 8.2e-112)
		tmp = Float64((2.0 ^ -1.0) * Float64(Float64(abs(p) + abs(r)) - Float64(r - p)));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 8.2e-112)
		tmp = (2.0 ^ -1.0) * ((abs(p) + abs(r)) - (r - p));
	else
		tmp = -q_m;
	end
	tmp_2 = 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[q$95$m, 8.2e-112], N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - N[(r - p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 8.19999999999999991e-112

   1. Initial program 18.9%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 11.7

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

   5. Applied rewrites 11.7%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 12.9%

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

   if 8.19999999999999991e-112 < q

   1. Initial program 22.2%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 8.2 \cdot 10^{-112}:\\ \;\;\;\;{2}^{-1} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \left(r - p\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{\left|r\right| + \left(\left|p\right| + p\right)}{r} \cdot 0.5 - 0.5\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= q_m 8.5e-14)
   (* (- (* (/ (+ (fabs r) (+ (fabs p) p)) r) 0.5) 0.5) r)
   (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 8.5e-14) {
		tmp = ((((fabs(r) + (fabs(p) + p)) / r) * 0.5) - 0.5) * r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 8.5d-14) then
        tmp = ((((abs(r) + (abs(p) + p)) / r) * 0.5d0) - 0.5d0) * r
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 8.5e-14) {
		tmp = ((((Math.abs(r) + (Math.abs(p) + p)) / r) * 0.5) - 0.5) * r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 8.5e-14:
		tmp = ((((math.fabs(r) + (math.fabs(p) + p)) / r) * 0.5) - 0.5) * r
	else:
		tmp = -q_m
	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 (q_m <= 8.5e-14)
		tmp = Float64(Float64(Float64(Float64(Float64(abs(r) + Float64(abs(p) + p)) / r) * 0.5) - 0.5) * r);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 8.5e-14)
		tmp = ((((abs(r) + (abs(p) + p)) / r) * 0.5) - 0.5) * r;
	else
		tmp = -q_m;
	end
	tmp_2 = 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[q$95$m, 8.5e-14], N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * r), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 8.50000000000000038e-14

   1. Initial program 19.6%

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

   3. Taylor expanded in r around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) \cdot 0.5} \]

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 5.1%

      \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]

   9. Taylor expanded in r around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 20.5%

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

   if 8.50000000000000038e-14 < q

   1. Initial program 20.7%

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

   3. Taylor expanded in 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 66.8

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

   5. Applied rewrites 66.8%

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

Alternative 4: 47.5% accurate, 16.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 8.2 \cdot 10^{-112}:\\ \;\;\;\;\left(0.5 - 0.5\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= q_m 8.2e-112) (* (- 0.5 0.5) r) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 8.2e-112) {
		tmp = (0.5 - 0.5) * r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 8.2d-112) then
        tmp = (0.5d0 - 0.5d0) * r
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 8.2e-112) {
		tmp = (0.5 - 0.5) * r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 8.2e-112:
		tmp = (0.5 - 0.5) * r
	else:
		tmp = -q_m
	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 (q_m <= 8.2e-112)
		tmp = Float64(Float64(0.5 - 0.5) * r);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 8.2e-112)
		tmp = (0.5 - 0.5) * r;
	else
		tmp = -q_m;
	end
	tmp_2 = 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[q$95$m, 8.2e-112], N[(N[(0.5 - 0.5), $MachinePrecision] * r), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 8.19999999999999991e-112

   1. Initial program 18.9%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 8.8%

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in p around 0

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

   10. Applied rewrites 23.8%

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

   if 8.19999999999999991e-112 < q

   1. Initial program 22.2%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.6

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

   5. Applied rewrites 62.6%

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

Alternative 5: 35.2% accurate, 83.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ -q\_m \end{array} \]

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 (- q_m))

C

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

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = -q_m
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	return -q_m;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return -q_m

Julia

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

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = -q_m;
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_] := (-q$95$m)

Derivation

1. Initial program 19.9%

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

3. Taylor expanded in q around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 22.9

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

5. Applied rewrites 22.9%

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

Reproduce

herbie shell --seed 2024339 
(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)))))))