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

Percentage Accurate: 24.0% → 60.6%

Time: 9.5s

Alternatives: 4

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: 24.0% 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: 60.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -2.3 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(\left(\left|p\right| + p\right) + \left|r\right|\right) \cdot \frac{0.5}{r} - 0.5\right) \cdot r\\ \mathbf{elif}\;r \leq 5.5 \cdot 10^{-62}:\\ \;\;\;\;-q\_m\\ \mathbf{else}:\\ \;\;\;\;{2}^{-1} \cdot \mathsf{fma}\left(\frac{p + r}{r}, \left(-2 \cdot q\_m\right) \cdot \frac{q\_m}{r}, p + \left|p\right|\right)\\ \end{array} \end{array} \]

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 -2.3e-160)
   (* (- (* (+ (+ (fabs p) p) (fabs r)) (/ 0.5 r)) 0.5) r)
   (if (<= r 5.5e-62)
     (- q_m)
     (*
      (pow 2.0 -1.0)
      (fma (/ (+ p r) r) (* (* -2.0 q_m) (/ q_m r)) (+ p (fabs p)))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= -2.3e-160) {
		tmp = ((((fabs(p) + p) + fabs(r)) * (0.5 / r)) - 0.5) * r;
	} else if (r <= 5.5e-62) {
		tmp = -q_m;
	} else {
		tmp = pow(2.0, -1.0) * fma(((p + r) / r), ((-2.0 * q_m) * (q_m / r)), (p + fabs(p)));
	}
	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 <= -2.3e-160)
		tmp = Float64(Float64(Float64(Float64(Float64(abs(p) + p) + abs(r)) * Float64(0.5 / r)) - 0.5) * r);
	elseif (r <= 5.5e-62)
		tmp = Float64(-q_m);
	else
		tmp = Float64((2.0 ^ -1.0) * fma(Float64(Float64(p + r) / r), Float64(Float64(-2.0 * q_m) * Float64(q_m / r)), Float64(p + abs(p))));
	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, -2.3e-160], N[(N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(0.5 / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[r, 5.5e-62], (-q$95$m), N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(N[(N[(p + r), $MachinePrecision] / r), $MachinePrecision] * N[(N[(-2.0 * q$95$m), $MachinePrecision] * N[(q$95$m / r), $MachinePrecision]), $MachinePrecision] + N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -2.29999999999999985e-160

   1. Initial program 15.7%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r} - \frac{1}{2}\right)} \]
   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 4.2%

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

   7. Applied rewrites 13.6%

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

   if -2.29999999999999985e-160 < r < 5.50000000000000022e-62

   1. Initial program 34.7%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 20.4

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

   5. Applied rewrites 20.4%

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

   if 5.50000000000000022e-62 < r

   1. Initial program 15.7%

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

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(r \cdot \left(\left(-2 \cdot \frac{p \cdot {q}^{2}}{{r}^{3}} + \left(\frac{\left|p\right|}{r} + \frac{\left|r\right|}{r}\right)\right) - \left(1 + \left(-1 \cdot \frac{p}{r} + 2 \cdot \frac{{q}^{2}}{{r}^{2}}\right)\right)\right)\right)} \]

   5. Applied rewrites 16.9%

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(q \cdot q\right) \cdot \frac{p}{{r}^{3}}, -2, \frac{\left|r\right| + \left|p\right|}{r} - \left(1 - \frac{\mathsf{fma}\left(\frac{q \cdot q}{r}, -2, p\right)}{r}\right)\right) \cdot r\right)} \]

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 26.7%

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

   10. Applied rewrites 31.7%

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

   12. Applied rewrites 59.0%

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\frac{p + r}{r}, \left(-2 \cdot q\right) \cdot \color{blue}{\frac{q}{r}}, \left(\left(p + \left(0 + \left|p\right|\right)\right) \cdot 1\right) \cdot 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.3 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(\left(\left|p\right| + p\right) + \left|r\right|\right) \cdot \frac{0.5}{r} - 0.5\right) \cdot r\\ \mathbf{elif}\;r \leq 5.5 \cdot 10^{-62}:\\ \;\;\;\;-q\\ \mathbf{else}:\\ \;\;\;\;{2}^{-1} \cdot \mathsf{fma}\left(\frac{p + r}{r}, \left(-2 \cdot q\right) \cdot \frac{q}{r}, p + \left|p\right|\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;4 \cdot {q\_m}^{2} \leq 2 \cdot 10^{-108}:\\ \;\;\;\;0 \cdot r\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= (* 4.0 (pow q_m 2.0)) 2e-108) (* 0.0 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 ((4.0 * pow(q_m, 2.0)) <= 2e-108) {
		tmp = 0.0 * 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 ((4.0d0 * (q_m ** 2.0d0)) <= 2d-108) then
        tmp = 0.0d0 * 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 ((4.0 * Math.pow(q_m, 2.0)) <= 2e-108) {
		tmp = 0.0 * 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 (4.0 * math.pow(q_m, 2.0)) <= 2e-108:
		tmp = 0.0 * 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 (Float64(4.0 * (q_m ^ 2.0)) <= 2e-108)
		tmp = Float64(0.0 * 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 ((4.0 * (q_m ^ 2.0)) <= 2e-108)
		tmp = 0.0 * 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[N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2e-108], N[(0.0 * r), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 2.00000000000000008e-108

   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 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 11.5%

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

   7. Applied rewrites 12.8%

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

   8. Taylor expanded in r around -inf

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

   10. Applied rewrites 37.4%

       \[\leadsto 0 \cdot r \]

   if 2.00000000000000008e-108 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 22.6%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 23.4

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

   5. Applied rewrites 23.4%

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

Alternative 3: 57.4% accurate, 14.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.4 \cdot 10^{-35}:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 1.4e-35) (* (+ (fabs p) p) 0.5) (- 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 <= 1.4e-35) {
		tmp = (fabs(p) + p) * 0.5;
	} 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 <= 1.4d-35) then
        tmp = (abs(p) + p) * 0.5d0
    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 <= 1.4e-35) {
		tmp = (Math.abs(p) + p) * 0.5;
	} 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 <= 1.4e-35:
		tmp = (math.fabs(p) + p) * 0.5
	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 <= 1.4e-35)
		tmp = Float64(Float64(abs(p) + p) * 0.5);
	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 <= 1.4e-35)
		tmp = (abs(p) + p) * 0.5;
	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, 1.4e-35], N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 1.4e-35

   1. Initial program 22.3%

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

   3. Taylor expanded in 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 7.6%

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

   7. Applied rewrites 7.7%

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

   8. Taylor expanded in r around 0

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

   10. Applied rewrites 21.0%

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

   if 1.4e-35 < q

   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 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 52.8

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

   5. Applied rewrites 52.8%

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

Alternative 4: 35.9% 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 21.8%

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

3. Taylor expanded in 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 16.4

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

5. Applied rewrites 16.4%

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

Reproduce

herbie shell --seed 2024337 
(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)))))))