The quadratic formula (r2)

Percentage Accurate: 52.5% → 90.9%

Time: 9.9s

Alternatives: 8

Speedup: 2.5×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.5% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-227}:\\ \;\;\;\;\frac{2 \cdot c}{t\_0 - b}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{t\_0 + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* c a) (* b b)))))
   (if (<= b -2.2e+121)
     (/ c (- b))
     (if (<= b 6.8e-227)
       (/ (* 2.0 c) (- t_0 b))
       (if (<= b 2.7e+71) (/ (+ t_0 b) (* -2.0 a)) (- (/ c b) (/ b a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (c * a), (b * b)));
	double tmp;
	if (b <= -2.2e+121) {
		tmp = c / -b;
	} else if (b <= 6.8e-227) {
		tmp = (2.0 * c) / (t_0 - b);
	} else if (b <= 2.7e+71) {
		tmp = (t_0 + b) / (-2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))
	tmp = 0.0
	if (b <= -2.2e+121)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6.8e-227)
		tmp = Float64(Float64(2.0 * c) / Float64(t_0 - b));
	elseif (b <= 2.7e+71)
		tmp = Float64(Float64(t_0 + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.2e+121], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6.8e-227], N[(N[(2.0 * c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e+71], N[(N[(t$95$0 + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.20000000000000001e121

   1. Initial program 10.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -2.20000000000000001e121 < b < 6.79999999999999958e-227

   1. Initial program 63.2%

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

   4. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c \cdot a, 4, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if 6.79999999999999958e-227 < b < 2.69999999999999997e71

   1. Initial program 72.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 72.5

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

   4. Applied rewrites 72.5%

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

   6. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}} \]

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 72.5

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

   8. Applied rewrites 72.5%

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

   if 2.69999999999999997e71 < b

   1. Initial program 55.7%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \color{blue}{\left(\mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+121)
   (/ c (- b))
   (if (<= b 1.6e+27)
     (/ (* 2.0 c) (- (sqrt (fma -4.0 (* c a) (* b b))) b))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+121) {
		tmp = c / -b;
	} else if (b <= 1.6e+27) {
		tmp = (2.0 * c) / (sqrt(fma(-4.0, (c * a), (b * b))) - b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+121)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.6e+27)
		tmp = Float64(Float64(2.0 * c) / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+121], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.6e+27], N[(N[(2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.20000000000000001e121

   1. Initial program 10.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -2.20000000000000001e121 < b < 1.60000000000000008e27

   1. Initial program 66.2%

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

   4. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c \cdot a, 4, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 72.7

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

   7. Applied rewrites 72.7%

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

   if 1.60000000000000008e27 < b

   1. Initial program 59.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \color{blue}{\left(\mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 89.2

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

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-40}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e-40)
   (/ c (- b))
   (if (<= b 1.6e+27)
     (/ (+ (sqrt (* (* -4.0 a) c)) b) (* -2.0 a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-40) {
		tmp = c / -b;
	} else if (b <= 1.6e+27) {
		tmp = (sqrt(((-4.0 * a) * c)) + b) / (-2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	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) :: tmp
    if (b <= (-1.95d-40)) then
        tmp = c / -b
    else if (b <= 1.6d+27) then
        tmp = (sqrt((((-4.0d0) * a) * c)) + b) / ((-2.0d0) * a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-40) {
		tmp = c / -b;
	} else if (b <= 1.6e+27) {
		tmp = (Math.sqrt(((-4.0 * a) * c)) + b) / (-2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-40:
		tmp = c / -b
	elif b <= 1.6e+27:
		tmp = (math.sqrt(((-4.0 * a) * c)) + b) / (-2.0 * a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-40)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.6e+27)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.95e-40)
		tmp = c / -b;
	elseif (b <= 1.6e+27)
		tmp = (sqrt(((-4.0 * a) * c)) + b) / (-2.0 * a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-40], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.6e+27], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9499999999999999e-40

   1. Initial program 18.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -1.9499999999999999e-40 < b < 1.60000000000000008e27

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 75.1

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

   4. Applied rewrites 75.1%

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

   6. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}} \]

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 75.1

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

   8. Applied rewrites 75.1%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 67.4

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

   11. Applied rewrites 67.4%

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

   if 1.60000000000000008e27 < b

   1. Initial program 59.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \color{blue}{\left(\mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 89.2

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

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 68.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ c (- b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	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) :: tmp
    if (b <= (-5d-310)) then
        tmp = c / -b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = c / -b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = c / -b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(c / (-b)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 37.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 66.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \color{blue}{\left(\mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 59.4

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

   5. Applied rewrites 59.4%

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

Alternative 5: 68.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-304) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-304) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	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) :: tmp
    if (b <= (-2.05d-304)) then
        tmp = c / -b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-304) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-304:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-304)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.05e-304)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-304], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.05000000000000001e-304

   1. Initial program 36.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -2.05000000000000001e-304 < b

   1. Initial program 66.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\right)}}{a} \]

      4. lower-neg.f64 58.2

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

   5. Applied rewrites 58.2%

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

Alternative 6: 34.5% accurate, 3.6× 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(c / Float64(-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 52.2%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

      \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

   6. lower-neg.f64 31.9

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

5. Applied rewrites 31.9%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Alternative 7: 10.2% accurate, 4.2× 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(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 52.2%

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

4. Applied rewrites 35.0%

   \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)} \]

5. Taylor expanded in a around 0

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

   1. lower-/.f64 11.6

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

7. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
8. Add Preprocessing

Alternative 8: 2.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

4. Applied rewrites 35.0%

   \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)} \]

5. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{\frac{b}{a}} \]
6. Step-by-step derivation

   1. lower-/.f64 2.5

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

7. Applied rewrites 2.5%

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

Developer Target 1: 70.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	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) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024288 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))