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

Percentage Accurate: 44.9% → 80.0%

Time: 8.0s

Alternatives: 12

Speedup: 11.4×

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: 44.9% 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: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left(\left|r\right| + \left|p\right|\right) + \sqrt{{q}^{2} \cdot 4 + {\left(p - r\right)}^{2}}\\ \mathbf{if}\;t\_0 \leq 10^{+153}:\\ \;\;\;\;\frac{1}{2} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left(\left|p\right| - p\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0
         (+
          (+ (fabs r) (fabs p))
          (sqrt (+ (* (pow q 2.0) 4.0) (pow (- p r) 2.0))))))
   (if (<= t_0 1e+153)
     (* (/ 1.0 2.0) t_0)
     (* (+ (+ (fabs r) r) (- (fabs p) p)) 0.5))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = (fabs(r) + fabs(p)) + sqrt(((pow(q, 2.0) * 4.0) + pow((p - r), 2.0)));
	double tmp;
	if (t_0 <= 1e+153) {
		tmp = (1.0 / 2.0) * t_0;
	} else {
		tmp = ((fabs(r) + r) + (fabs(p) - p)) * 0.5;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (abs(r) + abs(p)) + sqrt((((q ** 2.0d0) * 4.0d0) + ((p - r) ** 2.0d0)))
    if (t_0 <= 1d+153) then
        tmp = (1.0d0 / 2.0d0) * t_0
    else
        tmp = ((abs(r) + r) + (abs(p) - p)) * 0.5d0
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double t_0 = (Math.abs(r) + Math.abs(p)) + Math.sqrt(((Math.pow(q, 2.0) * 4.0) + Math.pow((p - r), 2.0)));
	double tmp;
	if (t_0 <= 1e+153) {
		tmp = (1.0 / 2.0) * t_0;
	} else {
		tmp = ((Math.abs(r) + r) + (Math.abs(p) - p)) * 0.5;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	t_0 = (math.fabs(r) + math.fabs(p)) + math.sqrt(((math.pow(q, 2.0) * 4.0) + math.pow((p - r), 2.0)))
	tmp = 0
	if t_0 <= 1e+153:
		tmp = (1.0 / 2.0) * t_0
	else:
		tmp = ((math.fabs(r) + r) + (math.fabs(p) - p)) * 0.5
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = Float64(Float64(abs(r) + abs(p)) + sqrt(Float64(Float64((q ^ 2.0) * 4.0) + (Float64(p - r) ^ 2.0))))
	tmp = 0.0
	if (t_0 <= 1e+153)
		tmp = Float64(Float64(1.0 / 2.0) * t_0);
	else
		tmp = Float64(Float64(Float64(abs(r) + r) + Float64(abs(p) - p)) * 0.5);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	t_0 = (abs(r) + abs(p)) + sqrt((((q ^ 2.0) * 4.0) + ((p - r) ^ 2.0)));
	tmp = 0.0;
	if (t_0 <= 1e+153)
		tmp = (1.0 / 2.0) * t_0;
	else
		tmp = ((abs(r) + r) + (abs(p) - p)) * 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_] := Block[{t$95$0 = N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[Power[q, 2.0], $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+153], N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))))) < 1e153

   1. Initial program 98.4%

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

   if 1e153 < (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))))))

   1. Initial program 7.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 30.9

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

   5. Applied rewrites 30.9%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 35.1%

       \[\leadsto \left(\left(\left|p\right| - p\right) + \left(\left|r\right| + r\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|r\right| + \left|p\right|\right) + \sqrt{{q}^{2} \cdot 4 + {\left(p - r\right)}^{2}} \leq 10^{+153}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\left|r\right| + \left|p\right|\right) + \sqrt{{q}^{2} \cdot 4 + {\left(p - r\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left(\left|p\right| - p\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := {\left(p - r\right)}^{2}\\ t_1 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;t\_1 + \sqrt{{q}^{2} \cdot 4 + t\_0} \leq 10^{+153}:\\ \;\;\;\;\frac{0.5}{\frac{1}{t\_1 + \sqrt{\mathsf{fma}\left(4, q \cdot q, t\_0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left(\left|p\right| - p\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (pow (- p r) 2.0)) (t_1 (+ (fabs r) (fabs p))))
   (if (<= (+ t_1 (sqrt (+ (* (pow q 2.0) 4.0) t_0))) 1e+153)
     (/ 0.5 (/ 1.0 (+ t_1 (sqrt (fma 4.0 (* q q) t_0)))))
     (* (+ (+ (fabs r) r) (- (fabs p) p)) 0.5))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = pow((p - r), 2.0);
	double t_1 = fabs(r) + fabs(p);
	double tmp;
	if ((t_1 + sqrt(((pow(q, 2.0) * 4.0) + t_0))) <= 1e+153) {
		tmp = 0.5 / (1.0 / (t_1 + sqrt(fma(4.0, (q * q), t_0))));
	} else {
		tmp = ((fabs(r) + r) + (fabs(p) - p)) * 0.5;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = Float64(p - r) ^ 2.0
	t_1 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (Float64(t_1 + sqrt(Float64(Float64((q ^ 2.0) * 4.0) + t_0))) <= 1e+153)
		tmp = Float64(0.5 / Float64(1.0 / Float64(t_1 + sqrt(fma(4.0, Float64(q * q), t_0)))));
	else
		tmp = Float64(Float64(Float64(abs(r) + r) + Float64(abs(p) - p)) * 0.5);
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[Sqrt[N[(N[(N[Power[q, 2.0], $MachinePrecision] * 4.0), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+153], N[(0.5 / N[(1.0 / N[(t$95$1 + N[Sqrt[N[(4.0 * N[(q * q), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))))) < 1e153

   1. Initial program 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\sqrt{\mathsf{fma}\left(4, q \cdot q, {\left(p - r\right)}^{2}\right)} + \left(\left|r\right| + \left|p\right|\right)}}} \]

   if 1e153 < (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))))))

   1. Initial program 7.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 30.9

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

   5. Applied rewrites 30.9%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 35.1%

       \[\leadsto \left(\left(\left|p\right| - p\right) + \left(\left|r\right| + r\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left|r\right| + \left|p\right|\right) + \sqrt{{q}^{2} \cdot 4 + {\left(p - r\right)}^{2}} \leq 10^{+153}:\\ \;\;\;\;\frac{0.5}{\frac{1}{\left(\left|r\right| + \left|p\right|\right) + \sqrt{\mathsf{fma}\left(4, q \cdot q, {\left(p - r\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left(\left|p\right| - p\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 23.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-46}:\\ \;\;\;\;\left(\left|r\right| + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1 \cdot q\\ \end{array} \end{array} \]

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-46) (* (+ (fabs r) (fabs p)) 0.5) (* 1.0 q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (pow(q, 2.0) <= 1e-46) {
		tmp = (fabs(r) + fabs(p)) * 0.5;
	} else {
		tmp = 1.0 * 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-46) then
        tmp = (abs(r) + abs(p)) * 0.5d0
    else
        tmp = 1.0d0 * 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-46) {
		tmp = (Math.abs(r) + Math.abs(p)) * 0.5;
	} else {
		tmp = 1.0 * 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-46:
		tmp = (math.fabs(r) + math.fabs(p)) * 0.5
	else:
		tmp = 1.0 * q
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if ((q ^ 2.0) <= 1e-46)
		tmp = Float64(Float64(abs(r) + abs(p)) * 0.5);
	else
		tmp = Float64(1.0 * 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-46)
		tmp = (abs(r) + abs(p)) * 0.5;
	else
		tmp = 1.0 * 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-46], N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(1.0 * q), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000002e-46

   1. Initial program 59.5%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 14.5

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

   5. Applied rewrites 14.5%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 18.2%

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

   if 1.00000000000000002e-46 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 37.3%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 34.8

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in q around inf

      \[\leadsto 1 \cdot q \]
   7. Step-by-step derivation

   8. Applied rewrites 26.5%

      \[\leadsto 1 \cdot q \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-46}:\\ \;\;\;\;\left(\left|r\right| + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1 \cdot q\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 1.8 \cdot 10^{+200}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left(\left|p\right| - p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{q} \cdot p, p, \mathsf{fma}\left(q, 2, \left|r\right| + \left|p\right|\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 1.8e+200)
   (* (+ (+ (fabs r) r) (- (fabs p) p)) 0.5)
   (fma (* (/ 0.125 q) p) p (* (fma q 2.0 (+ (fabs r) (fabs p))) 0.5))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 1.8e+200) {
		tmp = ((fabs(r) + r) + (fabs(p) - p)) * 0.5;
	} else {
		tmp = fma(((0.125 / q) * p), p, (fma(q, 2.0, (fabs(r) + fabs(p))) * 0.5));
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 1.8e+200)
		tmp = Float64(Float64(Float64(abs(r) + r) + Float64(abs(p) - p)) * 0.5);
	else
		tmp = fma(Float64(Float64(0.125 / q) * p), p, Float64(fma(q, 2.0, Float64(abs(r) + abs(p))) * 0.5));
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[q, 1.8e+200], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.125 / q), $MachinePrecision] * p), $MachinePrecision] * p + N[(N[(q * 2.0 + N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.7999999999999999e200

   1. Initial program 50.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 35.1%

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

   10. Applied rewrites 35.3%

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

   if 1.7999999999999999e200 < q

   1. Initial program 8.0%

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

   3. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 8.0

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

   5. Applied rewrites 8.0%

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

   6. Taylor expanded in p around 0

      \[\leadsto \frac{1}{8} \cdot \frac{{p}^{2}}{q} + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + 2 \cdot q\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 84.0%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{q} \cdot p, \color{blue}{p}, \mathsf{fma}\left(q, 2, \left|r\right| + \left|p\right|\right) \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 1.8 \cdot 10^{+200}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left(\left|p\right| - p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{q} \cdot p, p, \mathsf{fma}\left(q, 2, \left|r\right| + \left|p\right|\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;r \leq -3.35 \cdot 10^{-251}:\\ \;\;\;\;\left(t\_0 - p\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 1.02 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, q\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= r -3.35e-251)
     (* (- t_0 p) 0.5)
     (if (<= r 1.02e+30)
       (fma 0.5 t_0 q)
       (* (+ (+ (fabs r) r) (fabs p)) 0.5)))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (r <= -3.35e-251) {
		tmp = (t_0 - p) * 0.5;
	} else if (r <= 1.02e+30) {
		tmp = fma(0.5, t_0, q);
	} else {
		tmp = ((fabs(r) + r) + fabs(p)) * 0.5;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (r <= -3.35e-251)
		tmp = Float64(Float64(t_0 - p) * 0.5);
	elseif (r <= 1.02e+30)
		tmp = fma(0.5, t_0, q);
	else
		tmp = Float64(Float64(Float64(abs(r) + r) + abs(p)) * 0.5);
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, -3.35e-251], N[(N[(t$95$0 - p), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 1.02e+30], N[(0.5 * t$95$0 + q), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < -3.34999999999999989e-251

   1. Initial program 43.1%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 12.6

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

   5. Applied rewrites 12.6%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 23.4%

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

   if -3.34999999999999989e-251 < r < 1.02e30

   1. Initial program 65.0%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 33.4

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

   5. Applied rewrites 33.4%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 34.1%

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

   if 1.02e30 < r

   1. Initial program 28.9%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 75.9%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 71.4%

       \[\leadsto \left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 1.8 \cdot 10^{+200}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left(\left|p\right| - p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 1.8e+200)
   (* (+ (+ (fabs r) r) (- (fabs p) p)) 0.5)
   (fma 0.5 (+ (fabs r) (fabs p)) q)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 1.8e+200) {
		tmp = ((fabs(r) + r) + (fabs(p) - p)) * 0.5;
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q);
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 1.8e+200)
		tmp = Float64(Float64(Float64(abs(r) + r) + Float64(abs(p) - p)) * 0.5);
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q);
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[q, 1.8e+200], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.7999999999999999e200

   1. Initial program 50.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 35.1%

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

   10. Applied rewrites 35.3%

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

   if 1.7999999999999999e200 < q

   1. Initial program 8.0%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 84.8%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 1.8 \cdot 10^{+200}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left(\left|p\right| - p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.02 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r 1.02e+30)
   (fma 0.5 (+ (fabs r) (fabs p)) q)
   (* (+ (+ (fabs r) r) (fabs p)) 0.5)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 1.02e+30) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q);
	} else {
		tmp = ((fabs(r) + r) + fabs(p)) * 0.5;
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 1.02e+30)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q);
	else
		tmp = Float64(Float64(Float64(abs(r) + r) + abs(p)) * 0.5);
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[r, 1.02e+30], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.02e30

   1. Initial program 50.9%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 27.6

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

   5. Applied rewrites 27.6%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 29.2%

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

   if 1.02e30 < r

   1. Initial program 28.9%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 75.9%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 71.4%

       \[\leadsto \left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 28.8% accurate, 17.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\right) \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (fma 0.5 (+ (fabs r) (fabs p)) q))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	return fma(0.5, (fabs(r) + fabs(p)), q);
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	return fma(0.5, Float64(abs(r) + abs(p)), q)
end

Wolfram

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

Derivation

1. Initial program 46.1%

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

3. Taylor expanded in q around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-fabs.f64 N/A

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

   10. lower-fabs.f64 26.8

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

5. Applied rewrites 26.8%

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

6. Taylor expanded in q around 0

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

8. Applied rewrites 28.6%

   \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]
9. Add Preprocessing

Alternative 9: 20.2% accurate, 20.8× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.75 \cdot 10^{+169}:\\ \;\;\;\;1 \cdot q\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (if (<= r 1.75e+169) (* 1.0 q) (* 0.5 r)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 1.75e+169) {
		tmp = 1.0 * q;
	} else {
		tmp = 0.5 * r;
	}
	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 (r <= 1.75d+169) then
        tmp = 1.0d0 * q
    else
        tmp = 0.5d0 * r
    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 (r <= 1.75e+169) {
		tmp = 1.0 * q;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 1.75e+169:
		tmp = 1.0 * q
	else:
		tmp = 0.5 * r
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 1.75e+169)
		tmp = Float64(1.0 * q);
	else
		tmp = Float64(0.5 * r);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 1.75e+169)
		tmp = 1.0 * q;
	else
		tmp = 0.5 * r;
	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[r, 1.75e+169], N[(1.0 * q), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.75000000000000009e169

   1. Initial program 51.3%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 28.2

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in q around inf

      \[\leadsto 1 \cdot q \]
   7. Step-by-step derivation

   8. Applied rewrites 19.3%

      \[\leadsto 1 \cdot q \]

   if 1.75000000000000009e169 < r

   1. Initial program 7.8%

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

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 16.5

         \[\leadsto \color{blue}{0.5 \cdot r} \]

   5. Applied rewrites 16.5%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 12.9% accurate, 20.8× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 3.15 \cdot 10^{+29}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (if (<= r 3.15e+29) (* -0.5 p) (* 0.5 r)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 3.15e+29) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	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 (r <= 3.15d+29) then
        tmp = (-0.5d0) * p
    else
        tmp = 0.5d0 * r
    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 (r <= 3.15e+29) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 3.15e+29:
		tmp = -0.5 * p
	else:
		tmp = 0.5 * r
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 3.15e+29)
		tmp = Float64(-0.5 * p);
	else
		tmp = Float64(0.5 * r);
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 3.15e+29)
		tmp = -0.5 * p;
	else
		tmp = 0.5 * r;
	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[r, 3.15e+29], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.1499999999999999e29

   1. Initial program 50.9%

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

   3. Taylor expanded in p around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot p} \]
   4. Step-by-step derivation

      1. lower-*.f64 5.2

         \[\leadsto \color{blue}{-0.5 \cdot p} \]

   5. Applied rewrites 5.2%

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

   if 3.1499999999999999e29 < r

   1. Initial program 28.9%

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

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 14.9

         \[\leadsto \color{blue}{0.5 \cdot r} \]

   5. Applied rewrites 14.9%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 8.6% accurate, 41.7× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ -0.5 \cdot p \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (* -0.5 p))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	return -0.5 * p;
}

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 = (-0.5d0) * p
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	return -0.5 * p;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	return -0.5 * p

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	return Float64(-0.5 * p)
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp = code(p, r, q)
	tmp = -0.5 * p;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := N[(-0.5 * p), $MachinePrecision]

Derivation

1. Initial program 46.1%

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

3. Taylor expanded in p around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot p} \]
4. Step-by-step derivation

   1. lower-*.f64 4.8

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

5. Applied rewrites 4.8%

   \[\leadsto \color{blue}{-0.5 \cdot p} \]
6. Add Preprocessing

Alternative 12: 18.6% 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 46.1%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 18.1

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

5. Applied rewrites 18.1%

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

Reproduce

herbie shell --seed 2024295 
(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)))))))