quadp (p42, positive)

Percentage Accurate: 52.9% → 84.8%

Time: 14.4s

Alternatives: 6

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% 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: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -24000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -24000000.0)
   (- (/ c b) (/ b a))
   (if (<= b 1.12e-60)
     (/ (- (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 <= -24000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.12e-60) {
		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 <= (-24000000.0d0)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.12d-60) 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 <= -24000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.12e-60) {
		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 <= -24000000.0:
		tmp = (c / b) - (b / a)
	elif b <= 1.12e-60:
		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 <= -24000000.0)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.12e-60)
		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 <= -24000000.0)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.12e-60)
		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, -24000000.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.12e-60], 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 < -2.4e7

   1. Initial program 64.8%

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

      1. *-commutative 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in b around -inf 94.5%

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

      1. +-commutative 94.5%

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

      2. mul-1-neg 94.5%

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

      3. unsub-neg 94.5%

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

   7. Simplified 94.5%

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

   if -2.4e7 < b < 1.12e-60

   1. Initial program 72.8%

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

   if 1.12e-60 < b

   1. Initial program 21.0%

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

      1. *-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in b around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -24000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-61}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e-54)
   (- (/ c b) (/ b a))
   (if (<= b 7e-61) (* (- b (sqrt (* c (* a -4.0)))) (/ -0.5 a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e-54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7e-61) {
		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 <= (-3.1d-54)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7d-61) 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 <= -3.1e-54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7e-61) {
		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 <= -3.1e-54:
		tmp = (c / b) - (b / a)
	elif b <= 7e-61:
		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 <= -3.1e-54)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7e-61)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(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 <= -3.1e-54)
		tmp = (c / b) - (b / a);
	elseif (b <= 7e-61)
		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, -3.1e-54], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-61], N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.10000000000000004e-54

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in b around -inf 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

   7. Simplified 92.2%

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

   if -3.10000000000000004e-54 < b < 7.0000000000000006e-61

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in b around 0 63.9%

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

      1. associate-*r* 63.9%

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

   7. Simplified 63.9%

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

      1. clear-num 63.8%

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

      2. inv-pow 63.8%

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

   9. Applied egg-rr 63.8%

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

       1. unpow-1 63.8%

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

       2. associate-*r* 63.8%

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

   11. Simplified 63.8%

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

       1. frac-2neg 63.8%

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

       2. metadata-eval 63.8%

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

       3. div-inv 63.8%

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

       4. distribute-neg-frac 63.8%

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

       5. distribute-rgt-neg-in 63.8%

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

       6. metadata-eval 63.8%

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

       7. add-sqr-sqrt 34.6%

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

       8. sqrt-unprod 17.1%

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

       9. swap-sqr 17.1%

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

       10. metadata-eval 17.1%

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

       11. metadata-eval 17.1%

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

       12. swap-sqr 17.1%

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

       13. sqrt-unprod 1.0%

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

       14. add-sqr-sqrt 2.3%

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

       15. clear-num 2.3%

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

       16. div-inv 2.3%

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

   13. Applied egg-rr 63.9%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(b - \sqrt{c \cdot \left(-4 \cdot a\right)}\right) \cdot \frac{0.5}{a}\right)} \]
   14. Step-by-step derivation

       1. neg-mul-1 63.9%

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

       2. distribute-rgt-neg-in 63.9%

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

       3. *-commutative 63.9%

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

       4. distribute-neg-frac 63.9%

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

       5. metadata-eval 63.9%

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

   15. Simplified 63.9%

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

   if 7.0000000000000006e-61 < b

   1. Initial program 21.0%

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

      1. *-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in b around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-61}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-54)
   (- (/ c b) (/ b a))
   (if (<= b 1.45e-60)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.45e-60) {
		tmp = (sqrt((a * (c * -4.0))) - 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 <= (-4.6d-54)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.45d-60) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - 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 <= -4.6e-54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.45e-60) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e-54:
		tmp = (c / b) - (b / a)
	elif b <= 1.45e-60:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e-54)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.45e-60)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - 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 <= -4.6e-54)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.45e-60)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e-54], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-60], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $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 < -4.5999999999999998e-54

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in b around -inf 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

   7. Simplified 92.2%

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

   if -4.5999999999999998e-54 < b < 1.45e-60

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

   3. Simplified 68.3%

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

      1. add-sqr-sqrt 67.8%

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

      2. pow2 67.8%

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

      3. pow1/2 67.8%

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

      4. sqrt-pow1 67.8%

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

      5. fma-neg 67.8%

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

      6. distribute-lft-neg-in 67.8%

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

      7. *-commutative 67.8%

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

      8. associate-*r* 67.8%

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

      9. metadata-eval 67.8%

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

      10. metadata-eval 67.8%

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

   6. Applied egg-rr 67.8%

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

   7. Taylor expanded in c around inf 34.4%

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

      1. unpow2 34.4%

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

      2. fma-neg 34.4%

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

   9. Simplified 63.9%

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

   if 1.45e-60 < b

   1. Initial program 21.0%

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

      1. *-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in b around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 9e+14) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e+14) {
		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 <= 9d+14) 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 <= 9e+14) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9e+14:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9e+14)
		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 <= 9e+14)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9e+14], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9e14

   1. Initial program 66.9%

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

      1. *-commutative 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in b around -inf 49.6%

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

      1. associate-*r/ 49.6%

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

      2. mul-1-neg 49.6%

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

   7. Simplified 49.6%

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

   if 9e14 < b

   1. Initial program 20.7%

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

      1. *-commutative 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in b around inf 75.2%

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

      1. clear-num 74.3%

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

      2. inv-pow 74.3%

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

      3. *-commutative 74.3%

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

      4. times-frac 74.3%

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

      5. metadata-eval 74.3%

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

      6. associate-/l* 78.0%

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

   7. Applied egg-rr 78.0%

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

      1. unpow-1 78.0%

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

      2. mul-1-neg 78.0%

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

      3. associate-/r/ 87.3%

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

      4. *-inverses 87.3%

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

      5. *-lft-identity 87.3%

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

   9. Simplified 87.3%

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

       1. expm1-log1p-u 80.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{-\frac{b}{c}}\right)\right)} \]

       2. expm1-udef 34.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{-\frac{b}{c}}\right)} - 1} \]

       3. add-sqr-sqrt 16.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\sqrt{-\frac{b}{c}} \cdot \sqrt{-\frac{b}{c}}}}\right)} - 1 \]

       4. sqrt-unprod 32.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\sqrt{\left(-\frac{b}{c}\right) \cdot \left(-\frac{b}{c}\right)}}}\right)} - 1 \]

       5. sqr-neg 32.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\frac{b}{c} \cdot \frac{b}{c}}}}\right)} - 1 \]

       6. sqrt-unprod 17.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\sqrt{\frac{b}{c}} \cdot \sqrt{\frac{b}{c}}}}\right)} - 1 \]

       7. add-sqr-sqrt 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{b}{c}}}\right)} - 1 \]

       8. clear-num 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c}{b}}\right)} - 1 \]

   11. Applied egg-rr 29.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 29.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} \]

       2. expm1-log1p 29.1%

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

   13. Simplified 29.1%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 32.0%

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

      1. *-commutative 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in b around inf 66.3%

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

      1. associate-*r/ 66.3%

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

      2. neg-mul-1 66.3%

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

   7. Simplified 66.3%

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

4. Final simplification 66.7%

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

Alternative 6: 10.6% 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 52.3%

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

   1. *-commutative 52.3%

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

3. Simplified 52.3%

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

5. Taylor expanded in b around inf 28.9%

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

   1. clear-num 28.6%

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

   2. inv-pow 28.6%

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

   3. *-commutative 28.6%

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

   4. times-frac 28.6%

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

   5. metadata-eval 28.6%

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

   6. associate-/l* 29.9%

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

7. Applied egg-rr 29.9%

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

   1. unpow-1 29.9%

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

   2. mul-1-neg 29.9%

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

   3. associate-/r/ 33.9%

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

   4. *-inverses 33.9%

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

   5. *-lft-identity 33.9%

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

9. Simplified 33.9%

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

    1. expm1-log1p-u 30.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{-\frac{b}{c}}\right)\right)} \]

    2. expm1-udef 13.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{-\frac{b}{c}}\right)} - 1} \]

    3. add-sqr-sqrt 7.1%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\sqrt{-\frac{b}{c}} \cdot \sqrt{-\frac{b}{c}}}}\right)} - 1 \]

    4. sqrt-unprod 13.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\sqrt{\left(-\frac{b}{c}\right) \cdot \left(-\frac{b}{c}\right)}}}\right)} - 1 \]

    5. sqr-neg 13.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\frac{b}{c} \cdot \frac{b}{c}}}}\right)} - 1 \]

    6. sqrt-unprod 6.8%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\sqrt{\frac{b}{c}} \cdot \sqrt{\frac{b}{c}}}}\right)} - 1 \]

    7. add-sqr-sqrt 11.1%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{b}{c}}}\right)} - 1 \]

    8. clear-num 11.1%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c}{b}}\right)} - 1 \]

11. Applied egg-rr 11.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 10.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} \]

    2. expm1-log1p 11.3%

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

13. Simplified 11.3%

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

14. Final simplification 11.3%

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

Developer target: 99.7% accurate, 0.1× speedup

Math

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

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) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

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 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	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 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	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 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	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(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	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 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	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[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024031 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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