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

Percentage Accurate: 24.4% → 67.3%

Time: 8.3s

Alternatives: 8

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: 24.4% 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: 67.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-7}:\\ \;\;\;\;\left(\left(p + \left|p\right|\right) + \left(\left|r\right| - r\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{p}{q\_m}, 0.5, \frac{r}{q\_m} \cdot -0.25\right), r, \left|r\right| + \left|p\right|\right) - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 1e-7)
   (* (+ (+ p (fabs p)) (- (fabs r) r)) 0.5)
   (*
    (-
     (fma (fma (/ p q_m) 0.5 (* (/ r q_m) -0.25)) r (+ (fabs r) (fabs p)))
     (* q_m 2.0))
    0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e-7) {
		tmp = ((p + fabs(p)) + (fabs(r) - r)) * 0.5;
	} else {
		tmp = (fma(fma((p / q_m), 0.5, ((r / q_m) * -0.25)), r, (fabs(r) + fabs(p))) - (q_m * 2.0)) * 0.5;
	}
	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 ^ 2.0) <= 1e-7)
		tmp = Float64(Float64(Float64(p + abs(p)) + Float64(abs(r) - r)) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(Float64(p / q_m), 0.5, Float64(Float64(r / q_m) * -0.25)), r, Float64(abs(r) + abs(p))) - Float64(q_m * 2.0)) * 0.5);
	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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-7], N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(p / q$95$m), $MachinePrecision] * 0.5 + N[(N[(r / q$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * r + N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.9999999999999995e-8

   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 p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 8.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 38.5%

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

   if 9.9999999999999995e-8 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.9%

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

   3. Taylor expanded in p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.5%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 31.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{p}{q}, 0.5, \frac{r}{q} \cdot -0.25\right), r, \left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-7}:\\ \;\;\;\;\left(\left(p + \left|p\right|\right) + \left(\left|r\right| - r\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{r}{q\_m} \cdot -0.25, r, \left|r\right| + \left|p\right|\right) - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 1e-7)
   (* (+ (+ p (fabs p)) (- (fabs r) r)) 0.5)
   (* (- (fma (* (/ r q_m) -0.25) r (+ (fabs r) (fabs p))) (* q_m 2.0)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e-7) {
		tmp = ((p + fabs(p)) + (fabs(r) - r)) * 0.5;
	} else {
		tmp = (fma(((r / q_m) * -0.25), r, (fabs(r) + fabs(p))) - (q_m * 2.0)) * 0.5;
	}
	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 ^ 2.0) <= 1e-7)
		tmp = Float64(Float64(Float64(p + abs(p)) + Float64(abs(r) - r)) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(Float64(r / q_m) * -0.25), r, Float64(abs(r) + abs(p))) - Float64(q_m * 2.0)) * 0.5);
	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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-7], N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(r / q$95$m), $MachinePrecision] * -0.25), $MachinePrecision] * r + N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.9999999999999995e-8

   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 p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 8.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 38.5%

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

   if 9.9999999999999995e-8 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.9%

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

   3. Taylor expanded in p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.5%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 31.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{p}{q}, 0.5, \frac{r}{q} \cdot -0.25\right), r, \left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 31.7%

       \[\leadsto \left(\mathsf{fma}\left(\frac{r}{q} \cdot -0.25, r, \left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-7}:\\ \;\;\;\;\left(\left(p + \left|p\right|\right) + \left(\left|r\right| - r\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{q\_m}, 0.5, -1\right) \cdot q\_m\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 1e-7)
   (* (+ (+ p (fabs p)) (- (fabs r) r)) 0.5)
   (* (fma (/ (+ (fabs r) (fabs p)) q_m) 0.5 -1.0) 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 (pow(q_m, 2.0) <= 1e-7) {
		tmp = ((p + fabs(p)) + (fabs(r) - r)) * 0.5;
	} else {
		tmp = fma(((fabs(r) + fabs(p)) / q_m), 0.5, -1.0) * 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 ^ 2.0) <= 1e-7)
		tmp = Float64(Float64(Float64(p + abs(p)) + Float64(abs(r) - r)) * 0.5);
	else
		tmp = Float64(fma(Float64(Float64(abs(r) + abs(p)) / q_m), 0.5, -1.0) * q_m);
	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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-7], N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / q$95$m), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision] * q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.9999999999999995e-8

   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 p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 8.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 38.5%

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

   if 9.9999999999999995e-8 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.9%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 31.5

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

   5. Applied rewrites 31.5%

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

Alternative 4: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-7}:\\ \;\;\;\;\left(\left(p + \left|p\right|\right) + \left(\left|r\right| - r\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, q\_m, \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 1e-7)
   (* (+ (+ p (fabs p)) (- (fabs r) r)) 0.5)
   (* (+ (fma -2.0 q_m (fabs r)) (fabs p)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e-7) {
		tmp = ((p + fabs(p)) + (fabs(r) - r)) * 0.5;
	} else {
		tmp = (fma(-2.0, q_m, fabs(r)) + fabs(p)) * 0.5;
	}
	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 ^ 2.0) <= 1e-7)
		tmp = Float64(Float64(Float64(p + abs(p)) + Float64(abs(r) - r)) * 0.5);
	else
		tmp = Float64(Float64(fma(-2.0, q_m, abs(r)) + abs(p)) * 0.5);
	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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-7], N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(-2.0 * q$95$m + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.9999999999999995e-8

   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 p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 8.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 38.5%

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

   if 9.9999999999999995e-8 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - \mathsf{Rewrite=>}\left(lower-*.f64, \left(q \cdot 2\right)\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in r around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 27.5%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 31.5%

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

Alternative 5: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-7}:\\ \;\;\;\;\left(\left(p + \left|p\right|\right) + \left(\left|r\right| - r\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 1e-7)
   (* (+ (+ p (fabs p)) (- (fabs r) r)) 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 (pow(q_m, 2.0) <= 1e-7) {
		tmp = ((p + fabs(p)) + (fabs(r) - r)) * 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 ** 2.0d0) <= 1d-7) then
        tmp = ((p + abs(p)) + (abs(r) - r)) * 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 (Math.pow(q_m, 2.0) <= 1e-7) {
		tmp = ((p + Math.abs(p)) + (Math.abs(r) - r)) * 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 math.pow(q_m, 2.0) <= 1e-7:
		tmp = ((p + math.fabs(p)) + (math.fabs(r) - r)) * 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 ^ 2.0) <= 1e-7)
		tmp = Float64(Float64(Float64(p + abs(p)) + Float64(abs(r) - r)) * 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 ^ 2.0) <= 1e-7)
		tmp = ((p + abs(p)) + (abs(r) - r)) * 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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-7], N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.9999999999999995e-8

   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 p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 8.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 38.5%

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

   if 9.9999999999999995e-8 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

Alternative 6: 49.0% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.06 \cdot 10^{-13}:\\ \;\;\;\;\frac{q\_m \cdot q\_m}{-r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 1.06e-13) (/ (* q_m q_m) (- 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 <= 1.06e-13) {
		tmp = (q_m * q_m) / -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 <= 1.06d-13) then
        tmp = (q_m * q_m) / -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 <= 1.06e-13) {
		tmp = (q_m * q_m) / -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 <= 1.06e-13:
		tmp = (q_m * q_m) / -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 <= 1.06e-13)
		tmp = Float64(Float64(q_m * q_m) / Float64(-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 <= 1.06e-13)
		tmp = (q_m * q_m) / -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, 1.06e-13], N[(N[(q$95$m * q$95$m), $MachinePrecision] / (-r)), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 1.06e-13

   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 r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 10.6%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 26.4%

      \[\leadsto \frac{\left(-q\right) \cdot q}{\color{blue}{r}} \]

   if 1.06e-13 < q

   1. Initial program 30.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 63.0

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

   5. Applied rewrites 63.0%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 1.06 \cdot 10^{-13}:\\ \;\;\;\;\frac{q \cdot q}{-r}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.2% accurate, 11.4× speedup

Math

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

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) (* (+ (+ p (fabs r)) (fabs 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 = ((p + fabs(r)) + fabs(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 = ((p + abs(r)) + abs(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 = ((p + Math.abs(r)) + Math.abs(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 = ((p + math.fabs(r)) + math.fabs(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(Float64(p + abs(r)) + abs(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 = ((p + abs(r)) + abs(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[(p + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $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 p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 14.2%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 27.3%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 10.1%

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

   if 1.4e-35 < q

   1. Initial program 29.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

Alternative 8: 35.7% 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 24.2%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 20.7

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

5. Applied rewrites 20.7%

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

Reproduce

herbie shell --seed 2024340 
(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)))))))