quadm (p42, negative)

Percentage Accurate: 51.1% → 85.2%

Time: 8.2s

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: 51.1% 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: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 10^{+142}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-65)
   (/ c (- b))
   (if (<= b 1e+142)
     (/ (- (- b) (sqrt (/ 1.0 (/ 1.0 (fma -4.0 (* c a) (* b b)))))) (* 2.0 a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-65) {
		tmp = c / -b;
	} else if (b <= 1e+142) {
		tmp = (-b - sqrt((1.0 / (1.0 / fma(-4.0, (c * a), (b * b)))))) / (2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-65)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1e+142)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(1.0 / Float64(1.0 / fma(-4.0, Float64(c * a), Float64(b * b)))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-65], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1e+142], N[(N[((-b) - N[Sqrt[N[(1.0 / N[(1.0 / N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 < -3.7e-65

   1. Initial program 14.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   if -3.7e-65 < b < 1.00000000000000005e142

   1. Initial program 79.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 79.9

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 79.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 79.9

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

   4. Applied rewrites 79.9%

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

   if 1.00000000000000005e142 < b

   1. Initial program 50.1%

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

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 10^{+142}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-65)
   (/ c (- b))
   (if (<= b 1e+142)
     (/ (+ (sqrt (fma (* c a) -4.0 (* b b))) b) (* -2.0 a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-65) {
		tmp = c / -b;
	} else if (b <= 1e+142) {
		tmp = (sqrt(fma((c * a), -4.0, (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-65)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1e+142)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-65], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1e+142], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $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 < -3.7e-65

   1. Initial program 14.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   if -3.7e-65 < b < 1.00000000000000005e142

   1. Initial program 79.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 79.9

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 79.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 79.9

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

   4. Applied rewrites 79.9%

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

   6. Applied rewrites 79.9%

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

   if 1.00000000000000005e142 < b

   1. Initial program 50.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 10^{+142}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-65)
   (/ c (- b))
   (if (<= b 5.1e-45)
     (/ (+ (sqrt (* (* c a) -4.0)) b) (* -2.0 a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-65) {
		tmp = c / -b;
	} else if (b <= 5.1e-45) {
		tmp = (sqrt(((c * a) * -4.0)) + 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 <= (-3.7d-65)) then
        tmp = c / -b
    else if (b <= 5.1d-45) then
        tmp = (sqrt(((c * a) * (-4.0d0))) + 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 <= -3.7e-65) {
		tmp = c / -b;
	} else if (b <= 5.1e-45) {
		tmp = (Math.sqrt(((c * a) * -4.0)) + b) / (-2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-65:
		tmp = c / -b
	elif b <= 5.1e-45:
		tmp = (math.sqrt(((c * a) * -4.0)) + b) / (-2.0 * a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-65)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5.1e-45)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) + 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 <= -3.7e-65)
		tmp = c / -b;
	elseif (b <= 5.1e-45)
		tmp = (sqrt(((c * a) * -4.0)) + b) / (-2.0 * a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-65], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5.1e-45], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $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 < -3.7e-65

   1. Initial program 14.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   if -3.7e-65 < b < 5.0999999999999997e-45

   1. Initial program 70.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 70.5

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 70.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

   6. Applied rewrites 70.5%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 62.8

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

   9. Applied rewrites 62.8%

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

   if 5.0999999999999997e-45 < b

   1. Initial program 71.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(\sqrt{\left(c \cdot a\right) \cdot -4} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-65)
   (/ c (- b))
   (if (<= b 5.1e-45)
     (* (/ -0.5 a) (+ (sqrt (* (* c a) -4.0)) b))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-65) {
		tmp = c / -b;
	} else if (b <= 5.1e-45) {
		tmp = (-0.5 / a) * (sqrt(((c * a) * -4.0)) + 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 <= (-3.7d-65)) then
        tmp = c / -b
    else if (b <= 5.1d-45) then
        tmp = ((-0.5d0) / a) * (sqrt(((c * a) * (-4.0d0))) + 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 <= -3.7e-65) {
		tmp = c / -b;
	} else if (b <= 5.1e-45) {
		tmp = (-0.5 / a) * (Math.sqrt(((c * a) * -4.0)) + b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-65:
		tmp = c / -b
	elif b <= 5.1e-45:
		tmp = (-0.5 / a) * (math.sqrt(((c * a) * -4.0)) + b)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-65)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5.1e-45)
		tmp = Float64(Float64(-0.5 / a) * Float64(sqrt(Float64(Float64(c * a) * -4.0)) + 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 <= -3.7e-65)
		tmp = c / -b;
	elseif (b <= 5.1e-45)
		tmp = (-0.5 / a) * (sqrt(((c * a) * -4.0)) + b);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-65], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5.1e-45], N[(N[(-0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $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 < -3.7e-65

   1. Initial program 14.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   if -3.7e-65 < b < 5.0999999999999997e-45

   1. Initial program 70.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 70.5

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 70.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

   6. Applied rewrites 70.5%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 62.8

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

   9. Applied rewrites 62.8%

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

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. associate-/r/ N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-/r* N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. metadata-eval 62.7

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

   11. Applied rewrites 62.7%

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

   if 5.0999999999999997e-45 < b

   1. Initial program 71.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

Alternative 5: 68.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		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 <= (-5d-311)) 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 <= -5e-311) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		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 <= -5e-311)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 27.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 70.2

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

   5. Applied rewrites 70.2%

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

   if -5.00000000000023e-311 < b

   1. Initial program 73.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

Alternative 6: 35.4% 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 52.0%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
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 33.6

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

5. Applied rewrites 33.6%

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

Alternative 7: 11.6% 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 52.0%

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

4. Applied rewrites 29.6%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 12.4

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

7. Applied rewrites 12.4%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
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 52.0%

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

4. Applied rewrites 29.6%

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

5. Taylor expanded in b around -inf

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

   1. lower-/.f64 2.3

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

7. Applied rewrites 2.3%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024332 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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