quadp (p42, positive)

Percentage Accurate: 52.6% → 85.8%

Time: 11.7s

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: 52.6% 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: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+119}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+119)
   (/ b (- a))
   (if (<= b 1.15e-110)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+119) {
		tmp = b / -a;
	} else if (b <= 1.15e-110) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - 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 <= (-4d+119)) then
        tmp = b / -a
    else if (b <= 1.15d-110) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - 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 <= -4e+119) {
		tmp = b / -a;
	} else if (b <= 1.15e-110) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+119:
		tmp = b / -a
	elif b <= 1.15e-110:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+119)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.15e-110)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e+119)
		tmp = b / -a;
	elseif (b <= 1.15e-110)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+119], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.15e-110], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $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 < -3.99999999999999978e119

   1. Initial program 38.0%

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

      1. *-commutative 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in b around -inf 90.6%

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

      1. associate-*r/ 90.6%

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

      2. mul-1-neg 90.6%

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

   7. Simplified 90.6%

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

   if -3.99999999999999978e119 < b < 1.1500000000000001e-110

   1. Initial program 78.9%

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

   if 1.1500000000000001e-110 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+119}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-99)
   (/ b (- a))
   (if (<= b 8.5e-110)
     (* (/ 0.5 a) (- (sqrt (* c (* a -4.0))) b))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-99) {
		tmp = b / -a;
	} else if (b <= 8.5e-110) {
		tmp = (0.5 / a) * (sqrt((c * (a * -4.0))) - b);
	} 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 <= (-2.6d-99)) then
        tmp = b / -a
    else if (b <= 8.5d-110) then
        tmp = (0.5d0 / a) * (sqrt((c * (a * (-4.0d0)))) - b)
    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 <= -2.6e-99) {
		tmp = b / -a;
	} else if (b <= 8.5e-110) {
		tmp = (0.5 / a) * (Math.sqrt((c * (a * -4.0))) - b);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-99:
		tmp = b / -a
	elif b <= 8.5e-110:
		tmp = (0.5 / a) * (math.sqrt((c * (a * -4.0))) - b)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-99)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 8.5e-110)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(c * Float64(a * -4.0))) - b));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.6e-99)
		tmp = b / -a;
	elseif (b <= 8.5e-110)
		tmp = (0.5 / a) * (sqrt((c * (a * -4.0))) - b);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-99], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 8.5e-110], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.60000000000000005e-99

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in b around -inf 81.7%

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

      1. associate-*r/ 81.7%

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

      2. mul-1-neg 81.7%

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

   7. Simplified 81.7%

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

   if -2.60000000000000005e-99 < b < 8.50000000000000029e-110

   1. Initial program 66.8%

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

      1. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in b around 0 62.2%

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

      1. associate-*r* 62.2%

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

      2. *-commutative 62.2%

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

   7. Simplified 62.2%

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

      1. +-commutative 62.2%

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

      2. unsub-neg 62.2%

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

      3. *-commutative 62.2%

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

      4. associate-*r* 62.2%

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

   9. Applied egg-rr 62.2%

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

       1. div-sub 62.2%

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

       2. sub-neg 62.2%

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

       3. div-inv 62.3%

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

       4. *-commutative 62.3%

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

       5. *-commutative 62.3%

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

       6. associate-/r* 62.3%

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

       7. metadata-eval 62.3%

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

       8. div-inv 62.3%

          \[\leadsto \sqrt{\left(a \cdot c\right) \cdot -4} \cdot \frac{0.5}{a} + \left(-\color{blue}{b \cdot \frac{1}{a \cdot 2}}\right) \]

       9. *-commutative 62.3%

          \[\leadsto \sqrt{\left(a \cdot c\right) \cdot -4} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{1}{\color{blue}{2 \cdot a}}\right) \]

       10. associate-/r* 62.3%

           \[\leadsto \sqrt{\left(a \cdot c\right) \cdot -4} \cdot \frac{0.5}{a} + \left(-b \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\right) \]

       11. metadata-eval 62.3%

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

   11. Applied egg-rr 62.3%

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

       1. distribute-lft-neg-in 62.3%

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

       2. distribute-rgt-in 62.3%

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

       3. sub-neg 62.3%

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

       4. *-commutative 62.3%

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

       5. associate-*r* 62.3%

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

   13. Simplified 62.3%

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

   if 8.50000000000000029e-110 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-28}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 3e-28) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-28) {
		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 <= 3d-28) 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 <= 3e-28) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-28:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-28)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-28)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-28], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.00000000000000003e-28

   1. Initial program 65.8%

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

      1. *-commutative 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in b around -inf 47.9%

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

      1. associate-*r/ 47.9%

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

      2. mul-1-neg 47.9%

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

   7. Simplified 47.9%

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

   if 3.00000000000000003e-28 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

      1. clear-num 14.6%

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

      2. inv-pow 14.6%

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

   6. Applied egg-rr 7.4%

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

      1. unpow-1 7.4%

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

      2. associate-/l* 7.4%

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

      3. *-commutative 7.4%

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

   8. Simplified 7.4%

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

   9. Taylor expanded in b around -inf 34.3%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-28}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4e-308) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-308) {
		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 <= 2.4d-308) 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 <= 2.4e-308) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-308:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-308)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.4e-308)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-308], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000008e-308

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in b around -inf 64.1%

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

      1. associate-*r/ 64.1%

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

      2. mul-1-neg 64.1%

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

   7. Simplified 64.1%

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

   if 2.40000000000000008e-308 < b

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in b around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. neg-mul-1 72.3%

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

   7. Simplified 72.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.8%

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

   1. *-commutative 46.8%

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

3. Simplified 46.8%

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

   1. clear-num 46.7%

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

   2. inv-pow 46.7%

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

6. Applied egg-rr 25.5%

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

   1. unpow-1 25.5%

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

   2. associate-/l* 25.5%

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

   3. *-commutative 25.5%

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

8. Simplified 25.5%

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

9. Taylor expanded in a around 0 2.4%

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

10. Final simplification 2.4%

    \[\leadsto \frac{b}{a} \]
11. Add Preprocessing

Alternative 6: 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 46.8%

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

   1. *-commutative 46.8%

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

3. Simplified 46.8%

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

   1. clear-num 46.7%

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

   2. inv-pow 46.7%

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

6. Applied egg-rr 25.5%

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

   1. unpow-1 25.5%

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

   2. associate-/l* 25.5%

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

   3. *-commutative 25.5%

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

8. Simplified 25.5%

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

9. Taylor expanded in b around -inf 14.8%

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

10. Final simplification 14.8%

    \[\leadsto \frac{c}{b} \]
11. 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 2024076 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (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)))