Quadratic roots, medium range

Percentage Accurate: 32.2% → 95.4%

Time: 14.7s

Alternatives: 12

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 32.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{c \cdot {b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  a
  (-
   (*
    (pow c 4.0)
    (+ (* -5.0 (/ (pow a 2.0) (pow b 7.0))) (* -2.0 (/ a (* c (pow b 5.0))))))
   (/ (pow c 2.0) (pow b 3.0)))
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma(a, ((pow(c, 4.0) * ((-5.0 * (pow(a, 2.0) / pow(b, 7.0))) + (-2.0 * (a / (c * pow(b, 5.0)))))) - (pow(c, 2.0) / pow(b, 3.0))), (-c / b));
}

Julia

function code(a, b, c)
	return fma(a, Float64(Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64((a ^ 2.0) / (b ^ 7.0))) + Float64(-2.0 * Float64(a / Float64(c * (b ^ 5.0)))))) - Float64((c ^ 2.0) / (b ^ 3.0))), Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(a / N[(c * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 95.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 95.6%

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

   2. fma-define 95.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, -1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), -1 \cdot \frac{c}{b}\right)} \]

5. Simplified 95.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \frac{-0.25 \cdot \left(a \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right)}{b}\right) - \frac{{c}^{2}}{{b}^{3}}, \frac{c}{-b}\right)} \]

6. Taylor expanded in c around inf 95.6%

   \[\leadsto \mathsf{fma}\left(a, \color{blue}{{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5} \cdot c}\right)} - \frac{{c}^{2}}{{b}^{3}}, \frac{c}{-b}\right) \]

7. Final simplification 95.6%

   \[\leadsto \mathsf{fma}\left(a, {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{c \cdot {b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}, \frac{-c}{b}\right) \]
8. Add Preprocessing

Alternative 2: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}}\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (fma
   c
   (-
    (*
     c
     (fma
      -2.0
      (/ (pow a 2.0) (pow b 5.0))
      (* -5.0 (/ (* c (pow a 3.0)) (pow b 7.0)))))
    (/ a (pow b 3.0)))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * fma(c, ((c * fma(-2.0, (pow(a, 2.0) / pow(b, 5.0)), (-5.0 * ((c * pow(a, 3.0)) / pow(b, 7.0))))) - (a / pow(b, 3.0))), (-1.0 / b));
}

Julia

function code(a, b, c)
	return Float64(c * fma(c, Float64(Float64(c * fma(-2.0, Float64((a ^ 2.0) / (b ^ 5.0)), Float64(-5.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 7.0))))) - Float64(a / (b ^ 3.0))), Float64(-1.0 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(c * N[(N[(c * N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 95.4%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. fma-neg 95.4%

      \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right), -\frac{1}{b}\right)} \]

5. Simplified 95.4%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, \frac{-0.25}{a} \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{b}\right)\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right)} \]

6. Taylor expanded in a around 0 95.4%

   \[\leadsto c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right) \]

7. Final simplification 95.4%

   \[\leadsto c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}}\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right) \]
8. Add Preprocessing

Alternative 3: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -20.0)
   (/ (- (sqrt (fma a (* c -4.0) (pow b 2.0))) b) (* a 2.0))
   (- (/ (- c) b) (* a (/ (pow c 2.0) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -20.0) {
		tmp = (sqrt(fma(a, (c * -4.0), pow(b, 2.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -20.0)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), (b ^ 2.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -20.0], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -20

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 80.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      2. metadata-eval 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      3. *-commutative 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \color{blue}{\left(a \cdot c\right) \cdot -4}}}{2 \cdot a} \]

      4. associate-*r* 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \color{blue}{a \cdot \left(c \cdot -4\right)}}}{2 \cdot a} \]

      5. +-commutative 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + {b}^{2}}}}{2 \cdot a} \]

      6. fma-define 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}{2 \cdot a} \]

   5. Simplified 80.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}{2 \cdot a} \]

   if -20 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 26.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 94.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   4. Step-by-step derivation

      1. +-commutative 94.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]

      2. mul-1-neg 94.0%

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(-\frac{c}{b}\right)} \]

      3. unsub-neg 94.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} - \frac{c}{b}} \]

      4. mul-1-neg 94.0%

         \[\leadsto \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

      5. associate-/l* 94.0%

         \[\leadsto \left(-\color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}\right) - \frac{c}{b} \]

      6. distribute-rgt-neg-in 94.0%

         \[\leadsto \color{blue}{a \cdot \left(-\frac{{c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

      7. distribute-neg-frac2 94.0%

         \[\leadsto a \cdot \color{blue}{\frac{{c}^{2}}{-{b}^{3}}} - \frac{c}{b} \]

   5. Simplified 94.0%

      \[\leadsto \color{blue}{a \cdot \frac{{c}^{2}}{-{b}^{3}} - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (- (* -2.0 (/ (* a (pow c 3.0)) (pow b 5.0))) (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 5.0))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((-2.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((-2.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((-2.0 * ((a * (c ^ 3.0)) / (b ^ 5.0))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 94.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]

4. Final simplification 94.1%

   \[\leadsto a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -20.0) t_0 (- (/ (- c) b) (* a (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = t_0;
	} else {
		tmp = (-c / b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-20.0d0)) then
        tmp = t_0
    else
        tmp = (-c / b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = t_0;
	} else {
		tmp = (-c / b) - (a * (Math.pow(c, 2.0) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -20.0:
		tmp = t_0
	else:
		tmp = (-c / b) - (a * (math.pow(c, 2.0) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = t_0;
	else
		tmp = (-c / b) - (a * ((c ^ 2.0) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], t$95$0, N[(N[((-c) / b), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -20

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if -20 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 26.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 94.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   4. Step-by-step derivation

      1. +-commutative 94.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]

      2. mul-1-neg 94.0%

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(-\frac{c}{b}\right)} \]

      3. unsub-neg 94.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} - \frac{c}{b}} \]

      4. mul-1-neg 94.0%

         \[\leadsto \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

      5. associate-/l* 94.0%

         \[\leadsto \left(-\color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}\right) - \frac{c}{b} \]

      6. distribute-rgt-neg-in 94.0%

         \[\leadsto \color{blue}{a \cdot \left(-\frac{{c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

      7. distribute-neg-frac2 94.0%

         \[\leadsto a \cdot \color{blue}{\frac{{c}^{2}}{-{b}^{3}}} - \frac{c}{b} \]

   5. Simplified 94.0%

      \[\leadsto \color{blue}{a \cdot \frac{{c}^{2}}{-{b}^{3}} - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - a \cdot {b}^{-3}\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (* a (pow b -3.0))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a * pow(b, -3.0)))) + (-1.0 / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a * (b ** (-3.0d0))))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) - (a * Math.pow(b, -3.0)))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * ((-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) - (a * math.pow(b, -3.0)))) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a * (b ^ -3.0)))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-2.0 * ((c * (a ^ 2.0)) / (b ^ 5.0))) - (a * (b ^ -3.0)))) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 93.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. distribute-lft-in 93.9%

      \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right) + c \cdot \left(-1 \cdot \frac{a}{{b}^{3}}\right)\right)} - \frac{1}{b}\right) \]

   2. associate-/l* 93.9%

      \[\leadsto c \cdot \left(\left(c \cdot \left(-2 \cdot \color{blue}{\left({a}^{2} \cdot \frac{c}{{b}^{5}}\right)}\right) + c \cdot \left(-1 \cdot \frac{a}{{b}^{3}}\right)\right) - \frac{1}{b}\right) \]

   3. mul-1-neg 93.9%

      \[\leadsto c \cdot \left(\left(c \cdot \left(-2 \cdot \left({a}^{2} \cdot \frac{c}{{b}^{5}}\right)\right) + c \cdot \color{blue}{\left(-\frac{a}{{b}^{3}}\right)}\right) - \frac{1}{b}\right) \]

   4. div-inv 93.9%

      \[\leadsto c \cdot \left(\left(c \cdot \left(-2 \cdot \left({a}^{2} \cdot \frac{c}{{b}^{5}}\right)\right) + c \cdot \left(-\color{blue}{a \cdot \frac{1}{{b}^{3}}}\right)\right) - \frac{1}{b}\right) \]

   5. pow-flip 93.9%

      \[\leadsto c \cdot \left(\left(c \cdot \left(-2 \cdot \left({a}^{2} \cdot \frac{c}{{b}^{5}}\right)\right) + c \cdot \left(-a \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right) - \frac{1}{b}\right) \]

   6. metadata-eval 93.9%

      \[\leadsto c \cdot \left(\left(c \cdot \left(-2 \cdot \left({a}^{2} \cdot \frac{c}{{b}^{5}}\right)\right) + c \cdot \left(-a \cdot {b}^{\color{blue}{-3}}\right)\right) - \frac{1}{b}\right) \]

5. Applied egg-rr 93.9%

   \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(-2 \cdot \left({a}^{2} \cdot \frac{c}{{b}^{5}}\right)\right) + c \cdot \left(-a \cdot {b}^{-3}\right)\right)} - \frac{1}{b}\right) \]
6. Step-by-step derivation

   1. distribute-lft-out 93.9%

      \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(-2 \cdot \left({a}^{2} \cdot \frac{c}{{b}^{5}}\right) + \left(-a \cdot {b}^{-3}\right)\right)} - \frac{1}{b}\right) \]

   2. unsub-neg 93.9%

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \frac{c}{{b}^{5}}\right) - a \cdot {b}^{-3}\right)} - \frac{1}{b}\right) \]

   3. associate-*r/ 93.9%

      \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \color{blue}{\frac{{a}^{2} \cdot c}{{b}^{5}}} - a \cdot {b}^{-3}\right) - \frac{1}{b}\right) \]

   4. *-commutative 93.9%

      \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{\color{blue}{c \cdot {a}^{2}}}{{b}^{5}} - a \cdot {b}^{-3}\right) - \frac{1}{b}\right) \]

7. Simplified 93.9%

   \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - a \cdot {b}^{-3}\right)} - \frac{1}{b}\right) \]

8. Final simplification 93.9%

   \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - a \cdot {b}^{-3}\right) + \frac{-1}{b}\right) \]
9. Add Preprocessing

Alternative 7: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - \frac{\mathsf{fma}\left(2, {\left(a \cdot \frac{c}{b}\right)}^{2}, a \cdot c\right)}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (- (/ -1.0 b) (/ (fma 2.0 (pow (* a (/ c b)) 2.0) (* a c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (fma(2.0, pow((a * (c / b)), 2.0), (a * c)) / pow(b, 3.0)));
}

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(fma(2.0, (Float64(a * Float64(c / b)) ^ 2.0), Float64(a * c)) / (b ^ 3.0))))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(2.0 * N[Power[N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 93.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 89.3%

      \[\leadsto c \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right)\right)\right)} - \frac{1}{b}\right) \]

   2. expm1-undefine 84.5%

      \[\leadsto c \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right)\right)} - 1\right)} - \frac{1}{b}\right) \]

   3. fma-define 84.5%

      \[\leadsto c \cdot \left(\left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot c}{{b}^{5}}, -1 \cdot \frac{a}{{b}^{3}}\right)}\right)} - 1\right) - \frac{1}{b}\right) \]

   4. associate-/l* 84.5%

      \[\leadsto c \cdot \left(\left(e^{\mathsf{log1p}\left(c \cdot \mathsf{fma}\left(-2, \color{blue}{{a}^{2} \cdot \frac{c}{{b}^{5}}}, -1 \cdot \frac{a}{{b}^{3}}\right)\right)} - 1\right) - \frac{1}{b}\right) \]

   5. mul-1-neg 84.5%

      \[\leadsto c \cdot \left(\left(e^{\mathsf{log1p}\left(c \cdot \mathsf{fma}\left(-2, {a}^{2} \cdot \frac{c}{{b}^{5}}, \color{blue}{-\frac{a}{{b}^{3}}}\right)\right)} - 1\right) - \frac{1}{b}\right) \]

   6. div-inv 84.5%

      \[\leadsto c \cdot \left(\left(e^{\mathsf{log1p}\left(c \cdot \mathsf{fma}\left(-2, {a}^{2} \cdot \frac{c}{{b}^{5}}, -\color{blue}{a \cdot \frac{1}{{b}^{3}}}\right)\right)} - 1\right) - \frac{1}{b}\right) \]

   7. pow-flip 84.5%

      \[\leadsto c \cdot \left(\left(e^{\mathsf{log1p}\left(c \cdot \mathsf{fma}\left(-2, {a}^{2} \cdot \frac{c}{{b}^{5}}, -a \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)} - 1\right) - \frac{1}{b}\right) \]

   8. metadata-eval 84.5%

      \[\leadsto c \cdot \left(\left(e^{\mathsf{log1p}\left(c \cdot \mathsf{fma}\left(-2, {a}^{2} \cdot \frac{c}{{b}^{5}}, -a \cdot {b}^{\color{blue}{-3}}\right)\right)} - 1\right) - \frac{1}{b}\right) \]

5. Applied egg-rr 84.5%

   \[\leadsto c \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot \mathsf{fma}\left(-2, {a}^{2} \cdot \frac{c}{{b}^{5}}, -a \cdot {b}^{-3}\right)\right)} - 1\right)} - \frac{1}{b}\right) \]

6. Taylor expanded in b around -inf 93.9%

   \[\leadsto c \cdot \left(\color{blue}{-1 \cdot \frac{2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + a \cdot c}{{b}^{3}}} - \frac{1}{b}\right) \]
7. Step-by-step derivation

   1. mul-1-neg 93.9%

      \[\leadsto c \cdot \left(\color{blue}{\left(-\frac{2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + a \cdot c}{{b}^{3}}\right)} - \frac{1}{b}\right) \]

   2. distribute-neg-frac2 93.9%

      \[\leadsto c \cdot \left(\color{blue}{\frac{2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + a \cdot c}{-{b}^{3}}} - \frac{1}{b}\right) \]

   3. fma-define 93.9%

      \[\leadsto c \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, a \cdot c\right)}}{-{b}^{3}} - \frac{1}{b}\right) \]

   4. associate-/l* 93.9%

      \[\leadsto c \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{{a}^{2} \cdot \frac{{c}^{2}}{{b}^{2}}}, a \cdot c\right)}{-{b}^{3}} - \frac{1}{b}\right) \]

   5. unpow2 93.9%

      \[\leadsto c \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{\left(a \cdot a\right)} \cdot \frac{{c}^{2}}{{b}^{2}}, a \cdot c\right)}{-{b}^{3}} - \frac{1}{b}\right) \]

   6. unpow2 93.9%

      \[\leadsto c \cdot \left(\frac{\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, a \cdot c\right)}{-{b}^{3}} - \frac{1}{b}\right) \]

   7. unpow2 93.9%

      \[\leadsto c \cdot \left(\frac{\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}, a \cdot c\right)}{-{b}^{3}} - \frac{1}{b}\right) \]

   8. times-frac 93.9%

      \[\leadsto c \cdot \left(\frac{\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}, a \cdot c\right)}{-{b}^{3}} - \frac{1}{b}\right) \]

   9. swap-sqr 93.9%

      \[\leadsto c \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot \left(a \cdot \frac{c}{b}\right)}, a \cdot c\right)}{-{b}^{3}} - \frac{1}{b}\right) \]

   10. unpow2 93.9%

       \[\leadsto c \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2}}, a \cdot c\right)}{-{b}^{3}} - \frac{1}{b}\right) \]

8. Simplified 93.9%

   \[\leadsto c \cdot \left(\color{blue}{\frac{\mathsf{fma}\left(2, {\left(a \cdot \frac{c}{b}\right)}^{2}, a \cdot c\right)}{-{b}^{3}}} - \frac{1}{b}\right) \]

9. Final simplification 93.9%

   \[\leadsto c \cdot \left(\frac{-1}{b} - \frac{\mathsf{fma}\left(2, {\left(a \cdot \frac{c}{b}\right)}^{2}, a \cdot c\right)}{{b}^{3}}\right) \]
10. Add Preprocessing

Alternative 8: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 - \frac{a \cdot c}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -20.0) t_0 (/ (* c (- -1.0 (/ (* a c) (pow b 2.0)))) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = t_0;
	} else {
		tmp = (c * (-1.0 - ((a * c) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-20.0d0)) then
        tmp = t_0
    else
        tmp = (c * ((-1.0d0) - ((a * c) / (b ** 2.0d0)))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = t_0;
	} else {
		tmp = (c * (-1.0 - ((a * c) / Math.pow(b, 2.0)))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -20.0:
		tmp = t_0
	else:
		tmp = (c * (-1.0 - ((a * c) / math.pow(b, 2.0)))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * Float64(-1.0 - Float64(Float64(a * c) / (b ^ 2.0)))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = t_0;
	else
		tmp = (c * (-1.0 - ((a * c) / (b ^ 2.0)))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], t$95$0, N[(N[(c * N[(-1.0 - N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -20

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if -20 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 26.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 94.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg 94.0%

         \[\leadsto \frac{-1 \cdot c + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      2. unsub-neg 94.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot c - \frac{a \cdot {c}^{2}}{{b}^{2}}}}{b} \]

      3. mul-1-neg 94.0%

         \[\leadsto \frac{\color{blue}{\left(-c\right)} - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]

      4. associate-/l* 94.0%

         \[\leadsto \frac{\left(-c\right) - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b} \]

   5. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}} \]

   6. Taylor expanded in c around 0 94.0%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -20:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 - \frac{a \cdot c}{{b}^{2}}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 - \frac{a \cdot c}{{b}^{2}}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (- -1.0 (/ (* a c) (pow b 2.0)))) b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 - ((a * c) / pow(b, 2.0)))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c * ((-1.0d0) - ((a * c) / (b ** 2.0d0)))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 - ((a * c) / Math.pow(b, 2.0)))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 - ((a * c) / math.pow(b, 2.0)))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 - Float64(Float64(a * c) / (b ^ 2.0)))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 - ((a * c) / (b ^ 2.0)))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 - N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 90.7%

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. mul-1-neg 90.7%

      \[\leadsto \frac{-1 \cdot c + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

   2. unsub-neg 90.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot c - \frac{a \cdot {c}^{2}}{{b}^{2}}}}{b} \]

   3. mul-1-neg 90.7%

      \[\leadsto \frac{\color{blue}{\left(-c\right)} - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]

   4. associate-/l* 90.7%

      \[\leadsto \frac{\left(-c\right) - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b} \]

5. Simplified 90.7%

   \[\leadsto \color{blue}{\frac{\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}}{b}} \]

6. Taylor expanded in c around 0 90.6%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}}{b} \]

7. Final simplification 90.6%

   \[\leadsto \frac{c \cdot \left(-1 - \frac{a \cdot c}{{b}^{2}}\right)}{b} \]
8. Add Preprocessing

Alternative 10: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (/ (* a c) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((a * c) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-1.0d0) / b) - ((a * c) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((a * c) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - ((a * c) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(a * c) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - ((a * c) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 90.5%

   \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. sub-neg 90.5%

      \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b}^{3}} + \left(-\frac{1}{b}\right)\right)} \]

   2. mul-1-neg 90.5%

      \[\leadsto c \cdot \left(\color{blue}{\left(-\frac{a \cdot c}{{b}^{3}}\right)} + \left(-\frac{1}{b}\right)\right) \]

   3. distribute-neg-frac2 90.5%

      \[\leadsto c \cdot \left(\color{blue}{\frac{a \cdot c}{-{b}^{3}}} + \left(-\frac{1}{b}\right)\right) \]

   4. distribute-neg-frac 90.5%

      \[\leadsto c \cdot \left(\frac{a \cdot c}{-{b}^{3}} + \color{blue}{\frac{-1}{b}}\right) \]

   5. metadata-eval 90.5%

      \[\leadsto c \cdot \left(\frac{a \cdot c}{-{b}^{3}} + \frac{\color{blue}{-1}}{b}\right) \]

5. Simplified 90.5%

   \[\leadsto \color{blue}{c \cdot \left(\frac{a \cdot c}{-{b}^{3}} + \frac{-1}{b}\right)} \]

6. Final simplification 90.5%

   \[\leadsto c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right) \]
7. Add Preprocessing

Alternative 11: 80.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (- c) b))

C

double code(double a, double b, double c) {
	return -c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -c / b
end function

Java

public static double code(double a, double b, double c) {
	return -c / b;
}

Python

def code(a, b, c):
	return -c / b

Julia

function code(a, b, c)
	return Float64(Float64(-c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = -c / b;
end

Wolfram

code[a_, b_, c_] := N[((-c) / b), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 80.9%

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

   1. mul-1-neg 80.9%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   2. distribute-neg-frac2 80.9%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

5. Simplified 80.9%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]

6. Final simplification 80.9%

   \[\leadsto \frac{-c}{b} \]
7. Add Preprocessing

Alternative 12: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 31.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\sqrt{\left(4 \cdot a\right) \cdot c} \cdot \sqrt{\left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

   2. difference-of-squares 31.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{\left(4 \cdot a\right) \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a} \]

   3. associate-*l* 31.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   4. sqrt-prod 31.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   5. metadata-eval 31.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{2} \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   6. associate-*l* 31.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}}{2 \cdot a} \]

   7. sqrt-prod 31.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right)}}{2 \cdot a} \]

   8. metadata-eval 31.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{2} \cdot \sqrt{a \cdot c}\right)}}{2 \cdot a} \]

4. Applied egg-rr 31.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}}{2 \cdot a} \]

5. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
6. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}\right)}{a}} \]

   2. distribute-rgt-out 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{a \cdot c} \cdot \left(-2 + 2\right)\right)}}{a} \]

   3. *-commutative 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\color{blue}{c \cdot a}} \cdot \left(-2 + 2\right)\right)}{a} \]

   4. metadata-eval 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{c \cdot a} \cdot \color{blue}{0}\right)}{a} \]

   5. mul0-rgt 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

   6. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

7. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))