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

Percentage Accurate: 24.0% → 69.1%

Time: 9.2s

Alternatives: 4

Speedup: 2.0×

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: 69.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-43}:\\ \;\;\;\;\left(\left(\left|r\right| - r\right) + \left(\left|p\right| + p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - \mathsf{hypot}\left(-2 \cdot q, p - r\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= (pow q 2.0) 1e-43)
   (* (+ (- (fabs r) r) (+ (fabs p) p)) 0.5)
   (* (- (+ (fabs r) (fabs p)) (hypot (* -2.0 q) (- p r))) 0.5)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (pow(q, 2.0) <= 1e-43) {
		tmp = ((fabs(r) - r) + (fabs(p) + p)) * 0.5;
	} else {
		tmp = ((fabs(r) + fabs(p)) - hypot((-2.0 * q), (p - r))) * 0.5;
	}
	return tmp;
}

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (Math.pow(q, 2.0) <= 1e-43) {
		tmp = ((Math.abs(r) - r) + (Math.abs(p) + p)) * 0.5;
	} else {
		tmp = ((Math.abs(r) + Math.abs(p)) - Math.hypot((-2.0 * q), (p - r))) * 0.5;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if math.pow(q, 2.0) <= 1e-43:
		tmp = ((math.fabs(r) - r) + (math.fabs(p) + p)) * 0.5
	else:
		tmp = ((math.fabs(r) + math.fabs(p)) - math.hypot((-2.0 * q), (p - r))) * 0.5
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if ((q ^ 2.0) <= 1e-43)
		tmp = Float64(Float64(Float64(abs(r) - r) + Float64(abs(p) + p)) * 0.5);
	else
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) - hypot(Float64(-2.0 * q), Float64(p - r))) * 0.5);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if ((q ^ 2.0) <= 1e-43)
		tmp = ((abs(r) - r) + (abs(p) + p)) * 0.5;
	else
		tmp = ((abs(r) + abs(p)) - hypot((-2.0 * q), (p - r))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[N[Power[q, 2.0], $MachinePrecision], 1e-43], N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(-2.0 * q), $MachinePrecision] ^ 2 + N[(p - r), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000008e-43

   1. Initial program 27.7%

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

   3. Taylor expanded in 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 11.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 8.1%

      \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 8.1%

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

   11. Taylor expanded in q around 0

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

   13. Applied rewrites 41.2%

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

   if 1.00000000000000008e-43 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 23.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. sqr-abs N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. sqr-abs N/A

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

      12. div-sub N/A

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

      13. associate-+r- N/A

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

   4. Applied rewrites 21.8%

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

   6. Applied rewrites 59.1%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-43}:\\ \;\;\;\;\left(\left(\left|r\right| - r\right) + \left(\left|p\right| + p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - \mathsf{hypot}\left(-2 \cdot q, p - r\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-43}:\\ \;\;\;\;\left(\left(\left|r\right| - r\right) + \left(\left|p\right| + p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= (pow q 2.0) 1e-43) (* (+ (- (fabs r) r) (+ (fabs p) p)) 0.5) (- q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (pow(q, 2.0) <= 1e-43) {
		tmp = ((fabs(r) - r) + (fabs(p) + p)) * 0.5;
	} else {
		tmp = -q;
	}
	return tmp;
}

Fortran

NOTE: p, r, and q should be sorted in increasing order before calling this function.
real(8) function code(p, r, q)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if ((q ** 2.0d0) <= 1d-43) then
        tmp = ((abs(r) - r) + (abs(p) + p)) * 0.5d0
    else
        tmp = -q
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (Math.pow(q, 2.0) <= 1e-43) {
		tmp = ((Math.abs(r) - r) + (Math.abs(p) + p)) * 0.5;
	} else {
		tmp = -q;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if math.pow(q, 2.0) <= 1e-43:
		tmp = ((math.fabs(r) - r) + (math.fabs(p) + p)) * 0.5
	else:
		tmp = -q
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if ((q ^ 2.0) <= 1e-43)
		tmp = Float64(Float64(Float64(abs(r) - r) + Float64(abs(p) + p)) * 0.5);
	else
		tmp = Float64(-q);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if ((q ^ 2.0) <= 1e-43)
		tmp = ((abs(r) - r) + (abs(p) + p)) * 0.5;
	else
		tmp = -q;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[N[Power[q, 2.0], $MachinePrecision], 1e-43], N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-q)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000008e-43

   1. Initial program 27.7%

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

   3. Taylor expanded in 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 11.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 8.1%

      \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 8.1%

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

   11. Taylor expanded in q around 0

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

   13. Applied rewrites 41.2%

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

   if 1.00000000000000008e-43 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 23.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 29.8

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

   5. Applied rewrites 29.8%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-43}:\\ \;\;\;\;\left(\left(\left|r\right| - r\right) + \left(\left|p\right| + p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 34.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 20000000:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= (pow q 2.0) 20000000.0) (/ (* (- q) q) r) (- q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (pow(q, 2.0) <= 20000000.0) {
		tmp = (-q * q) / r;
	} else {
		tmp = -q;
	}
	return tmp;
}

Fortran

NOTE: p, r, and q should be sorted in increasing order before calling this function.
real(8) function code(p, r, q)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if ((q ** 2.0d0) <= 20000000.0d0) then
        tmp = (-q * q) / r
    else
        tmp = -q
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (Math.pow(q, 2.0) <= 20000000.0) {
		tmp = (-q * q) / r;
	} else {
		tmp = -q;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if math.pow(q, 2.0) <= 20000000.0:
		tmp = (-q * q) / r
	else:
		tmp = -q
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if ((q ^ 2.0) <= 20000000.0)
		tmp = Float64(Float64(Float64(-q) * q) / r);
	else
		tmp = Float64(-q);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if ((q ^ 2.0) <= 20000000.0)
		tmp = (-q * q) / r;
	else
		tmp = -q;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[N[Power[q, 2.0], $MachinePrecision], 20000000.0], N[(N[((-q) * q), $MachinePrecision] / r), $MachinePrecision], (-q)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 2e7

   1. Initial program 28.6%

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

   4. Applied rewrites 7.5%

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left({r}^{3} + {p}^{3}, \frac{1}{\mathsf{fma}\left(r, r, p \cdot p\right) - \left|r \cdot p\right|}, -\sqrt{\mathsf{fma}\left(4, q \cdot q, {\left(p - r\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in r around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in q around inf

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

   10. Applied rewrites 38.6%

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

   if 2e7 < (pow.f64 q #s(literal 2 binary64))

   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 30.6

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

   5. Applied rewrites 30.6%

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

Alternative 4: 19.0% accurate, 83.3× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ -q \end{array} \]

FPCore

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

C

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

Fortran

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

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	return -q;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	return -q

Julia

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

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp = code(p, r, q)
	tmp = -q;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := (-q)

Derivation

1. Initial program 25.3%

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

3. Taylor expanded in 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 20.8

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

5. Applied rewrites 20.8%

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

Reproduce

herbie shell --seed 2024285 
(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)))))))