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

Percentage Accurate: 45.0% → 99.9%

Time: 7.6s

Alternatives: 9

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: 45.0% 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: 99.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \mathsf{hypot}\left(2 \cdot q, p - r\right)\right) \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (* 0.5 (+ (+ (fabs p) (fabs r)) (hypot (* 2.0 q) (- p r)))))

C

double code(double p, double r, double q) {
	return 0.5 * ((fabs(p) + fabs(r)) + hypot((2.0 * q), (p - r)));
}

Java

public static double code(double p, double r, double q) {
	return 0.5 * ((Math.abs(p) + Math.abs(r)) + Math.hypot((2.0 * q), (p - r)));
}

Python

def code(p, r, q):
	return 0.5 * ((math.fabs(p) + math.fabs(r)) + math.hypot((2.0 * q), (p - r)))

Julia

function code(p, r, q)
	return Float64(0.5 * Float64(Float64(abs(p) + abs(r)) + hypot(Float64(2.0 * q), Float64(p - r))))
end

MATLAB

function tmp = code(p, r, q)
	tmp = 0.5 * ((abs(p) + abs(r)) + hypot((2.0 * q), (p - r)));
end

Wolfram

code[p_, r_, q_] := N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(2.0 * q), $MachinePrecision] ^ 2 + N[(p - r), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 48.8

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 48.8

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

4. Applied rewrites 48.8%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 48.8

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

6. Applied rewrites 48.8%

   \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{\mathsf{fma}\left(p - r, p - r, 4 \cdot \left(q \cdot q\right)\right)}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 48.8

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

   \[\leadsto 0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \mathsf{hypot}\left(2 \cdot q, p - r\right)\right) \]
10. Add Preprocessing

Alternative 2: 43.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\left(\left(\left(\left|r\right| + r\right) + \left|p\right|\right) - p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(q, 2, \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (pow q 2.0) 5e+189)
   (* (- (+ (+ (fabs r) r) (fabs p)) p) 0.5)
   (* (+ (fma q 2.0 (fabs r)) (fabs p)) 0.5)))

C

double code(double p, double r, double q) {
	double tmp;
	if (pow(q, 2.0) <= 5e+189) {
		tmp = (((fabs(r) + r) + fabs(p)) - p) * 0.5;
	} else {
		tmp = (fma(q, 2.0, fabs(r)) + fabs(p)) * 0.5;
	}
	return tmp;
}

Julia

function code(p, r, q)
	tmp = 0.0
	if ((q ^ 2.0) <= 5e+189)
		tmp = Float64(Float64(Float64(Float64(abs(r) + r) + abs(p)) - p) * 0.5);
	else
		tmp = Float64(Float64(fma(q, 2.0, abs(r)) + abs(p)) * 0.5);
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Power[q, 2.0], $MachinePrecision], 5e+189], N[(N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - p), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(q * 2.0 + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 5.0000000000000004e189

   1. Initial program 57.2%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   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 36.1

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

   5. Applied rewrites 36.1%

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

   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 41.8%

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

   if 5.0000000000000004e189 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 26.7%

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

   3. Taylor expanded in r around 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 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 36.5%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\left(\left(\left(\left|r\right| + r\right) + \left|p\right|\right) - p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(q, 2, \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\left(\left(\left(\left|r\right| + r\right) + \left|p\right|\right) - p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot q\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (pow q 2.0) 5e+189)
   (* (- (+ (+ (fabs r) r) (fabs p)) p) 0.5)
   (* (* 2.0 q) 0.5)))

C

double code(double p, double r, double q) {
	double tmp;
	if (pow(q, 2.0) <= 5e+189) {
		tmp = (((fabs(r) + r) + fabs(p)) - p) * 0.5;
	} else {
		tmp = (2.0 * q) * 0.5;
	}
	return tmp;
}

Fortran

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) <= 5d+189) then
        tmp = (((abs(r) + r) + abs(p)) - p) * 0.5d0
    else
        tmp = (2.0d0 * q) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (Math.pow(q, 2.0) <= 5e+189) {
		tmp = (((Math.abs(r) + r) + Math.abs(p)) - p) * 0.5;
	} else {
		tmp = (2.0 * q) * 0.5;
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.pow(q, 2.0) <= 5e+189:
		tmp = (((math.fabs(r) + r) + math.fabs(p)) - p) * 0.5
	else:
		tmp = (2.0 * q) * 0.5
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if ((q ^ 2.0) <= 5e+189)
		tmp = Float64(Float64(Float64(Float64(abs(r) + r) + abs(p)) - p) * 0.5);
	else
		tmp = Float64(Float64(2.0 * q) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if ((q ^ 2.0) <= 5e+189)
		tmp = (((abs(r) + r) + abs(p)) - p) * 0.5;
	else
		tmp = (2.0 * q) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Power[q, 2.0], $MachinePrecision], 5e+189], N[(N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - p), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(2.0 * q), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 5.0000000000000004e189

   1. Initial program 57.2%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   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 36.1

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

   5. Applied rewrites 36.1%

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

   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 41.8%

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

   if 5.0000000000000004e189 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 26.7%

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

   3. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 11.4

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

   5. Applied rewrites 11.4%

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

   6. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.8

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

   8. Applied rewrites 31.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval 31.8

         \[\leadsto \left(q \cdot 2\right) \cdot \color{blue}{0.5} \]

   10. Applied rewrites 31.8%

       \[\leadsto \color{blue}{\left(q \cdot 2\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\left(\left(\left(\left|r\right| + r\right) + \left|p\right|\right) - p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot q\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -4.6 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(\left|p\right| + \left|r\right|\right) - p\right) \cdot 0.5\\ \mathbf{elif}\;p \leq -1.56 \cdot 10^{-261}:\\ \;\;\;\;\left(2 \cdot q\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= p -4.6e+25)
   (* (- (+ (fabs p) (fabs r)) p) 0.5)
   (if (<= p -1.56e-261)
     (* (* 2.0 q) 0.5)
     (* (+ (+ (fabs r) r) (fabs p)) 0.5))))

C

double code(double p, double r, double q) {
	double tmp;
	if (p <= -4.6e+25) {
		tmp = ((fabs(p) + fabs(r)) - p) * 0.5;
	} else if (p <= -1.56e-261) {
		tmp = (2.0 * q) * 0.5;
	} else {
		tmp = ((fabs(r) + r) + fabs(p)) * 0.5;
	}
	return tmp;
}

Fortran

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 (p <= (-4.6d+25)) then
        tmp = ((abs(p) + abs(r)) - p) * 0.5d0
    else if (p <= (-1.56d-261)) then
        tmp = (2.0d0 * q) * 0.5d0
    else
        tmp = ((abs(r) + r) + abs(p)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (p <= -4.6e+25) {
		tmp = ((Math.abs(p) + Math.abs(r)) - p) * 0.5;
	} else if (p <= -1.56e-261) {
		tmp = (2.0 * q) * 0.5;
	} else {
		tmp = ((Math.abs(r) + r) + Math.abs(p)) * 0.5;
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if p <= -4.6e+25:
		tmp = ((math.fabs(p) + math.fabs(r)) - p) * 0.5
	elif p <= -1.56e-261:
		tmp = (2.0 * q) * 0.5
	else:
		tmp = ((math.fabs(r) + r) + math.fabs(p)) * 0.5
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (p <= -4.6e+25)
		tmp = Float64(Float64(Float64(abs(p) + abs(r)) - p) * 0.5);
	elseif (p <= -1.56e-261)
		tmp = Float64(Float64(2.0 * q) * 0.5);
	else
		tmp = Float64(Float64(Float64(abs(r) + r) + abs(p)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (p <= -4.6e+25)
		tmp = ((abs(p) + abs(r)) - p) * 0.5;
	elseif (p <= -1.56e-261)
		tmp = (2.0 * q) * 0.5;
	else
		tmp = ((abs(r) + r) + abs(p)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[p, -4.6e+25], N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - p), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[p, -1.56e-261], N[(N[(2.0 * q), $MachinePrecision] * 0.5), $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 p < -4.5999999999999996e25

   1. Initial program 39.5%

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

   3. Taylor expanded in 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 52.7

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

   5. Applied rewrites 52.7%

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

   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 67.6%

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

   if -4.5999999999999996e25 < p < -1.55999999999999993e-261

   1. Initial program 67.7%

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

   3. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 15.8

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

   5. Applied rewrites 15.8%

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

   6. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 22.9

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

   8. Applied rewrites 22.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval 22.9

         \[\leadsto \left(q \cdot 2\right) \cdot \color{blue}{0.5} \]

   10. Applied rewrites 22.9%

       \[\leadsto \color{blue}{\left(q \cdot 2\right) \cdot 0.5} \]

   if -1.55999999999999993e-261 < p

   1. Initial program 42.6%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 16.7

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

   5. Applied rewrites 16.7%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 28.9%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -4.6 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(\left|p\right| + \left|r\right|\right) - p\right) \cdot 0.5\\ \mathbf{elif}\;p \leq -1.56 \cdot 10^{-261}:\\ \;\;\;\;\left(2 \cdot q\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 31.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 6.1 \cdot 10^{-65}:\\ \;\;\;\;\left(2 \cdot q\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= r 6.1e-65) (* (* 2.0 q) 0.5) (* (+ (+ (fabs r) r) (fabs p)) 0.5)))

C

double code(double p, double r, double q) {
	double tmp;
	if (r <= 6.1e-65) {
		tmp = (2.0 * q) * 0.5;
	} else {
		tmp = ((fabs(r) + r) + fabs(p)) * 0.5;
	}
	return tmp;
}

Fortran

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 <= 6.1d-65) then
        tmp = (2.0d0 * q) * 0.5d0
    else
        tmp = ((abs(r) + r) + abs(p)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (r <= 6.1e-65) {
		tmp = (2.0 * q) * 0.5;
	} else {
		tmp = ((Math.abs(r) + r) + Math.abs(p)) * 0.5;
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if r <= 6.1e-65:
		tmp = (2.0 * q) * 0.5
	else:
		tmp = ((math.fabs(r) + r) + math.fabs(p)) * 0.5
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (r <= 6.1e-65)
		tmp = Float64(Float64(2.0 * q) * 0.5);
	else
		tmp = Float64(Float64(Float64(abs(r) + r) + abs(p)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 6.1e-65)
		tmp = (2.0 * q) * 0.5;
	else
		tmp = ((abs(r) + r) + abs(p)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[r, 6.1e-65], N[(N[(2.0 * q), $MachinePrecision] * 0.5), $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 < 6.10000000000000014e-65

   1. Initial program 51.9%

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

   3. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 23.9

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

   5. Applied rewrites 23.9%

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

   6. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 14.0

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

   8. Applied rewrites 14.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval 14.0

         \[\leadsto \left(q \cdot 2\right) \cdot \color{blue}{0.5} \]

   10. Applied rewrites 14.0%

       \[\leadsto \color{blue}{\left(q \cdot 2\right) \cdot 0.5} \]

   if 6.10000000000000014e-65 < r

   1. Initial program 42.3%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 60.4%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.1 \cdot 10^{-65}:\\ \;\;\;\;\left(2 \cdot q\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 19.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot q\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= p -2e+154) (* -0.5 p) (* (* 2.0 q) 0.5)))

C

double code(double p, double r, double q) {
	double tmp;
	if (p <= -2e+154) {
		tmp = -0.5 * p;
	} else {
		tmp = (2.0 * q) * 0.5;
	}
	return tmp;
}

Fortran

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 (p <= (-2d+154)) then
        tmp = (-0.5d0) * p
    else
        tmp = (2.0d0 * q) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (p <= -2e+154) {
		tmp = -0.5 * p;
	} else {
		tmp = (2.0 * q) * 0.5;
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if p <= -2e+154:
		tmp = -0.5 * p
	else:
		tmp = (2.0 * q) * 0.5
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (p <= -2e+154)
		tmp = Float64(-0.5 * p);
	else
		tmp = Float64(Float64(2.0 * q) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (p <= -2e+154)
		tmp = -0.5 * p;
	else
		tmp = (2.0 * q) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[p, -2e+154], N[(-0.5 * p), $MachinePrecision], N[(N[(2.0 * q), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -2.00000000000000007e154

   1. Initial program 7.6%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 16.4

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

   5. Applied rewrites 16.4%

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

   if -2.00000000000000007e154 < p

   1. Initial program 53.2%

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

   3. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 17.2

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

   5. Applied rewrites 17.2%

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

   6. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 14.1

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

   8. Applied rewrites 14.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval 14.1

         \[\leadsto \left(q \cdot 2\right) \cdot \color{blue}{0.5} \]

   10. Applied rewrites 14.1%

       \[\leadsto \color{blue}{\left(q \cdot 2\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot q\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 7.8% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -5 \cdot 10^{-86}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (p r q) :precision binary64 (if (<= p -5e-86) (* -0.5 p) (* 0.5 r)))

C

double code(double p, double r, double q) {
	double tmp;
	if (p <= -5e-86) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Fortran

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 (p <= (-5d-86)) then
        tmp = (-0.5d0) * p
    else
        tmp = 0.5d0 * r
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (p <= -5e-86) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if p <= -5e-86:
		tmp = -0.5 * p
	else:
		tmp = 0.5 * r
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (p <= -5e-86)
		tmp = Float64(-0.5 * p);
	else
		tmp = Float64(0.5 * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (p <= -5e-86)
		tmp = -0.5 * p;
	else
		tmp = 0.5 * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[p, -5e-86], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -4.9999999999999999e-86

   1. Initial program 50.8%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 11.3

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

   5. Applied rewrites 11.3%

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

   if -4.9999999999999999e-86 < p

   1. Initial program 47.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 6.4

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

   5. Applied rewrites 6.4%

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

Alternative 8: 5.3% accurate, 41.7× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot p \end{array} \]

FPCore

(FPCore (p r q) :precision binary64 (* -0.5 p))

C

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

Fortran

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

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

Python

def code(p, r, q):
	return -0.5 * p

Julia

function code(p, r, q)
	return Float64(-0.5 * p)
end

MATLAB

function tmp = code(p, r, q)
	tmp = -0.5 * p;
end

Wolfram

code[p_, r_, q_] := N[(-0.5 * p), $MachinePrecision]

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in p around -inf

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

   1. lower-*.f64 5.0

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

5. Applied rewrites 5.0%

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

Alternative 9: 18.4% accurate, 83.3× speedup

Math

\[\begin{array}{l} \\ -q \end{array} \]

FPCore

(FPCore (p r q) :precision binary64 (- q))

C

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

Fortran

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

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

Python

def code(p, r, q):
	return -q

Julia

function code(p, r, q)
	return Float64(-q)
end

MATLAB

function tmp = code(p, r, q)
	tmp = -q;
end

Wolfram

code[p_, r_, q_] := (-q)

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in q around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 19.3

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

5. Applied rewrites 19.3%

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

Reproduce

herbie shell --seed 2024331 
(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)))))))