quadp (p42, positive)

Percentage Accurate: 51.4% → 85.0%

Time: 12.8s

Alternatives: 8

Speedup: 19.1×

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: 51.4% 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: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, \frac{1}{b}, \frac{-c}{\frac{\frac{{b}^{3}}{a}}{c}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (- (/ c b) (/ b a))
   (if (<= b 2.8e-65)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (fma (- c) (/ 1.0 b) (/ (- c) (/ (/ (pow b 3.0) a) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.8e-65) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = fma(-c, (1.0 / b), (-c / ((pow(b, 3.0) / a) / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.8e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = fma(Float64(-c), Float64(1.0 / b), Float64(Float64(-c) / Float64(Float64((b ^ 3.0) / a) / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-65], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) * N[(1.0 / b), $MachinePrecision] + N[((-c) / N[(N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 26.4%

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

      1. neg-sub0 26.4%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 26.4%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 26.4%

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

      4. neg-mul-1 26.4%

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

      5. *-commutative 26.4%

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

      6. associate-*r/ 26.4%

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

   3. Simplified 26.7%

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

   4. Taylor expanded in b around -inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. unsub-neg 97.7%

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

   6. Simplified 97.7%

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

   if -5.00000000000000018e153 < b < 2.8e-65

   1. Initial program 83.0%

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

   if 2.8e-65 < b

   1. Initial program 13.8%

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

      1. neg-sub0 13.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 13.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 13.8%

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

      4. neg-mul-1 13.8%

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

      5. *-commutative 13.8%

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

      6. associate-*r/ 13.8%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in b around inf 77.5%

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

      1. +-commutative 77.5%

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

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

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

      4. associate-*r/ 77.5%

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

      5. neg-mul-1 77.5%

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

      6. unpow2 77.5%

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

      7. associate-*l* 86.8%

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

   6. Simplified 86.8%

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

      1. div-inv 86.5%

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

      2. fma-neg 86.6%

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

      3. associate-/l* 91.0%

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

      4. *-commutative 91.0%

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

   8. Applied egg-rr 91.0%

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

      1. distribute-neg-frac 91.0%

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

      2. associate-/r* 93.1%

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

   10. Simplified 93.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, \frac{1}{b}, \frac{-c}{\frac{\frac{{b}^{3}}{a}}{c}}\right)\\ \end{array} \]

Alternative 2: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+84}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-92}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e+84)
   (- (/ c b) (/ b a))
   (if (<= b 1.12e-92)
     (* (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+84) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.12e-92) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = -c / 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) :: tmp
    if (b <= (-1.1d+84)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.12d-92) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) * (0.5d0 / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+84) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.12e-92) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.1e+84:
		tmp = (c / b) - (b / a)
	elif b <= 1.12e-92:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e+84)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.12e-92)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e+84)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.12e-92)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e+84], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.12e-92], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0999999999999999e84

   1. Initial program 45.5%

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

      1. neg-sub0 45.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 45.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 45.5%

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

      4. neg-mul-1 45.5%

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

      5. *-commutative 45.5%

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

      6. associate-*r/ 45.4%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in b around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

   6. Simplified 98.3%

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

   if -1.0999999999999999e84 < b < 1.11999999999999999e-92

   1. Initial program 81.9%

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

      1. /-rgt-identity 81.9%

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

      2. metadata-eval 81.9%

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

      3. associate-/l* 81.9%

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

      4. associate-*r/ 81.7%

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

      5. +-commutative 81.7%

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

      6. unsub-neg 81.7%

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

      7. fma-neg 81.7%

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

      8. *-commutative 81.7%

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

      9. distribute-rgt-neg-in 81.7%

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

      10. associate-*l* 80.8%

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

      11. metadata-eval 80.8%

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

      12. associate-/r* 80.8%

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

      13. metadata-eval 80.8%

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

      14. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

      1. fma-udef 80.8%

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

   5. Applied egg-rr 80.8%

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

   if 1.11999999999999999e-92 < b

   1. Initial program 15.4%

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

      1. neg-sub0 15.4%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 15.4%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 15.4%

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

      4. neg-mul-1 15.4%

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

      5. *-commutative 15.4%

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

      6. associate-*r/ 15.4%

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

   3. Simplified 15.4%

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

   4. Taylor expanded in b around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   6. Simplified 91.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+84}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-92}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (- (/ c b) (/ b a))
   (if (<= b 5.5e-72)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.5e-72) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -c / 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) :: tmp
    if (b <= (-5d+153)) then
        tmp = (c / b) - (b / a)
    else if (b <= 5.5d-72) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.5e-72) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+153:
		tmp = (c / b) - (b / a)
	elif b <= 5.5e-72:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.5e-72)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+153)
		tmp = (c / b) - (b / a);
	elseif (b <= 5.5e-72)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e-72], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 26.4%

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

      1. neg-sub0 26.4%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 26.4%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 26.4%

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

      4. neg-mul-1 26.4%

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

      5. *-commutative 26.4%

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

      6. associate-*r/ 26.4%

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

   3. Simplified 26.7%

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

   4. Taylor expanded in b around -inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. unsub-neg 97.7%

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

   6. Simplified 97.7%

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

   if -5.00000000000000018e153 < b < 5.49999999999999994e-72

   1. Initial program 83.0%

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

   if 5.49999999999999994e-72 < b

   1. Initial program 13.8%

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

      1. neg-sub0 13.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 13.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 13.8%

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

      4. neg-mul-1 13.8%

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

      5. *-commutative 13.8%

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

      6. associate-*r/ 13.8%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in b around inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. neg-mul-1 93.0%

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

   6. Simplified 93.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;\left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-54)
   (/ (* 2.0 (- (/ c (/ b a)) b)) (* a 2.0))
   (if (<= b 1.15e-92)
     (* (- b (sqrt (* (* c a) -4.0))) (/ -0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-54) {
		tmp = (2.0 * ((c / (b / a)) - b)) / (a * 2.0);
	} else if (b <= 1.15e-92) {
		tmp = (b - sqrt(((c * a) * -4.0))) * (-0.5 / a);
	} else {
		tmp = -c / 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) :: tmp
    if (b <= (-1.5d-54)) then
        tmp = (2.0d0 * ((c / (b / a)) - b)) / (a * 2.0d0)
    else if (b <= 1.15d-92) then
        tmp = (b - sqrt(((c * a) * (-4.0d0)))) * ((-0.5d0) / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-54) {
		tmp = (2.0 * ((c / (b / a)) - b)) / (a * 2.0);
	} else if (b <= 1.15e-92) {
		tmp = (b - Math.sqrt(((c * a) * -4.0))) * (-0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-54:
		tmp = (2.0 * ((c / (b / a)) - b)) / (a * 2.0)
	elif b <= 1.15e-92:
		tmp = (b - math.sqrt(((c * a) * -4.0))) * (-0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-54)
		tmp = Float64(Float64(2.0 * Float64(Float64(c / Float64(b / a)) - b)) / Float64(a * 2.0));
	elseif (b <= 1.15e-92)
		tmp = Float64(Float64(b - sqrt(Float64(Float64(c * a) * -4.0))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e-54)
		tmp = (2.0 * ((c / (b / a)) - b)) / (a * 2.0);
	elseif (b <= 1.15e-92)
		tmp = (b - sqrt(((c * a) * -4.0))) * (-0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-54], N[(N[(2.0 * N[(N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-92], N[(N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.50000000000000005e-54

   1. Initial program 64.2%

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

      1. add-sqr-sqrt 64.0%

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

      2. pow2 64.0%

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

      3. pow1/2 64.0%

         \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}\right)}^{2}}{2 \cdot a} \]

      4. sqrt-pow1 64.0%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{2 \cdot a} \]

      5. cancel-sign-sub-inv 64.0%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(b \cdot b + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      6. fma-def 64.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      7. metadata-eval 64.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      8. metadata-eval 64.2%

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

   3. Applied egg-rr 64.2%

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

   4. Taylor expanded in b around -inf 83.9%

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

      1. mul-1-neg 83.9%

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

      2. distribute-lft-out 83.9%

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

      3. associate-*l/ 92.0%

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

      4. *-commutative 92.0%

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

      5. unsub-neg 92.0%

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

      6. *-commutative 92.0%

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

      7. associate-/r/ 92.0%

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

   6. Simplified 92.0%

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

   if -1.50000000000000005e-54 < b < 1.15000000000000008e-92

   1. Initial program 75.5%

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

      1. neg-sub0 75.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 75.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 75.5%

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

      4. neg-mul-1 75.5%

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

      5. *-commutative 75.5%

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

      6. associate-*r/ 75.4%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in a around inf 73.8%

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

   if 1.15000000000000008e-92 < b

   1. Initial program 15.4%

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

      1. neg-sub0 15.4%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 15.4%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 15.4%

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

      4. neg-mul-1 15.4%

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

      5. *-commutative 15.4%

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

      6. associate-*r/ 15.4%

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

   3. Simplified 15.4%

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

   4. Taylor expanded in b around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   6. Simplified 91.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;\left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / 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) :: tmp
    if (b <= (-2d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 68.1%

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

      1. neg-sub0 68.1%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 68.1%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 68.1%

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

      4. neg-mul-1 68.1%

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

      5. *-commutative 68.1%

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

      6. associate-*r/ 67.9%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in b around -inf 65.5%

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

      1. mul-1-neg 65.5%

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

      2. unsub-neg 65.5%

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

   6. Simplified 65.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 30.3%

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

      1. neg-sub0 30.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 30.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 30.3%

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

      4. neg-mul-1 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*r/ 30.2%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in b around inf 73.1%

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

      1. associate-*r/ 73.1%

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

      2. neg-mul-1 73.1%

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

   6. Simplified 73.1%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 43.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 0.095) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.095) {
		tmp = -b / a;
	} else {
		tmp = c / 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) :: tmp
    if (b <= 0.095d0) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.095) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.095:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.095)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.095)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.095], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.095000000000000001

   1. Initial program 65.3%

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

      1. neg-sub0 65.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 65.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 65.3%

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

      4. neg-mul-1 65.3%

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

      5. *-commutative 65.3%

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

      6. associate-*r/ 65.1%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in b around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. mul-1-neg 47.2%

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

   6. Simplified 47.2%

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

   if 0.095000000000000001 < b

   1. Initial program 13.1%

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

      1. /-rgt-identity 13.1%

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

      2. metadata-eval 13.1%

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

      3. associate-/l* 13.1%

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

      4. associate-*r/ 13.1%

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

      5. +-commutative 13.1%

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

      6. unsub-neg 13.1%

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

      7. fma-neg 13.1%

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

      8. *-commutative 13.1%

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

      9. distribute-rgt-neg-in 13.1%

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

      10. associate-*l* 13.1%

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

      11. metadata-eval 13.1%

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

      12. associate-/r* 13.1%

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

      13. metadata-eval 13.1%

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

      14. metadata-eval 13.1%

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

   3. Simplified 13.1%

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

   4. Taylor expanded in b around -inf 2.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + -2 \cdot b\right)} \cdot \frac{0.5}{a} \]

   5. Taylor expanded in c around inf 32.5%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7: 67.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.7e-284) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.7e-284) {
		tmp = -b / a;
	} else {
		tmp = -c / 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) :: tmp
    if (b <= 4.7d-284) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.7e-284) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.7e-284:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.7e-284)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.7e-284)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.7e-284], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.70000000000000022e-284

   1. Initial program 67.8%

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

      1. neg-sub0 67.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 67.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 67.8%

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

      4. neg-mul-1 67.8%

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

      5. *-commutative 67.8%

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

      6. associate-*r/ 67.7%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in b around -inf 63.9%

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

      1. associate-*r/ 63.9%

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

      2. mul-1-neg 63.9%

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

   6. Simplified 63.9%

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

   if 4.70000000000000022e-284 < b

   1. Initial program 30.0%

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

      1. neg-sub0 30.0%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

      2. associate-+l- 30.0%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      3. sub0-neg 30.0%

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

      4. neg-mul-1 30.0%

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

      5. *-commutative 30.0%

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

      6. associate-*r/ 29.9%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in b around inf 74.1%

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

      1. associate-*r/ 74.1%

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

      2. neg-mul-1 74.1%

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

   6. Simplified 74.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 11.2% accurate, 38.7× 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 48.7%

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

   1. /-rgt-identity 48.7%

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

   2. metadata-eval 48.7%

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

   3. associate-/l* 48.7%

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

   4. associate-*r/ 48.6%

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

   5. +-commutative 48.6%

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

   6. unsub-neg 48.6%

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

   7. fma-neg 48.6%

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

   8. *-commutative 48.6%

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

   9. distribute-rgt-neg-in 48.6%

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

   10. associate-*l* 48.3%

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

   11. metadata-eval 48.3%

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

   12. associate-/r* 48.3%

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

   13. metadata-eval 48.3%

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

   14. metadata-eval 48.3%

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

3. Simplified 48.3%

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

4. Taylor expanded in b around -inf 30.4%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + -2 \cdot b\right)} \cdot \frac{0.5}{a} \]

5. Taylor expanded in c around inf 12.3%

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

6. Final simplification 12.3%

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

Developer target: 70.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* 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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-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(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * 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 = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-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[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023229 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

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