quadm (p42, negative)

Percentage Accurate: 52.2% → 86.5%

Time: 11.4s

Alternatives: 7

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: 52.2% 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: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+143}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.1e-90)
   (/ (- c) b)
   (if (<= b 9.5e+143)
     (* -0.5 (/ (+ b (sqrt (fma b b (* a (* c -4.0))))) a))
     (* -0.5 (/ (+ b (+ b (* -2.0 (* c (/ a b))))) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.1e-90) {
		tmp = -c / b;
	} else if (b <= 9.5e+143) {
		tmp = -0.5 * ((b + sqrt(fma(b, b, (a * (c * -4.0))))) / a);
	} else {
		tmp = -0.5 * ((b + (b + (-2.0 * (c * (a / b))))) / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.1e-90)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 9.5e+143)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(fma(b, b, Float64(a * Float64(c * -4.0))))) / a));
	else
		tmp = Float64(-0.5 * Float64(Float64(b + Float64(b + Float64(-2.0 * Float64(c * Float64(a / b))))) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.1e-90], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 9.5e+143], N[(-0.5 * N[(N[(b + N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.10000000000000035e-90

   1. Initial program 17.1%

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

   2. Taylor expanded in b around -inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   4. Simplified 82.3%

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

   if -4.10000000000000035e-90 < b < 9.50000000000000066e143

   1. Initial program 77.9%

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

      1. /-rgt-identity 77.9%

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

      2. metadata-eval 77.9%

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

      3. associate-/l* 77.7%

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

      4. associate-/r/ 77.7%

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

      5. *-commutative 77.7%

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

      6. metadata-eval 77.7%

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

      7. metadata-eval 77.7%

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

      8. associate-*l/ 77.7%

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

      9. associate-/r/ 77.7%

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

      10. times-frac 77.9%

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

      11. *-commutative 77.9%

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

      12. times-frac 77.9%

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

      13. metadata-eval 77.9%

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

      14. associate-/r/ 77.9%

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

      15. *-commutative 77.9%

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

      16. div-sub 77.9%

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

   3. Simplified 77.9%

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

   if 9.50000000000000066e143 < b

   1. Initial program 49.0%

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

      1. /-rgt-identity 49.0%

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

      2. metadata-eval 49.0%

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

      3. associate-/l* 49.0%

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

      4. associate-/r/ 49.0%

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

      5. *-commutative 49.0%

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

      6. metadata-eval 49.0%

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

      7. metadata-eval 49.0%

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

      8. associate-*l/ 49.0%

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

      9. associate-/r/ 49.0%

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

      10. times-frac 49.0%

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

      11. *-commutative 49.0%

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

      12. times-frac 49.0%

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

      13. metadata-eval 49.0%

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

      14. associate-/r/ 49.0%

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

      15. *-commutative 49.0%

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

      16. div-sub 49.0%

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

   3. Simplified 49.0%

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

      1. fma-udef 49.0%

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

   5. Applied egg-rr 49.0%

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

   6. Taylor expanded in b around inf 87.0%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}{a} \]
   7. Step-by-step derivation

      1. associate-*r/ 100.0%

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

   8. Simplified 100.0%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+143}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}{a}\\ \end{array} \]

Alternative 2: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-90)
   (/ (- c) b)
   (if (<= b 1.5e+142)
     (* -0.5 (/ (+ b (sqrt (+ (* a (* c -4.0)) (* b b)))) a))
     (* -0.5 (/ (+ b (+ b (* -2.0 (* c (/ a b))))) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-90) {
		tmp = -c / b;
	} else if (b <= 1.5e+142) {
		tmp = -0.5 * ((b + sqrt(((a * (c * -4.0)) + (b * b)))) / a);
	} else {
		tmp = -0.5 * ((b + (b + (-2.0 * (c * (a / b))))) / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.6d-90)) then
        tmp = -c / b
    else if (b <= 1.5d+142) then
        tmp = (-0.5d0) * ((b + sqrt(((a * (c * (-4.0d0))) + (b * b)))) / a)
    else
        tmp = (-0.5d0) * ((b + (b + ((-2.0d0) * (c * (a / b))))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-90) {
		tmp = -c / b;
	} else if (b <= 1.5e+142) {
		tmp = -0.5 * ((b + Math.sqrt(((a * (c * -4.0)) + (b * b)))) / a);
	} else {
		tmp = -0.5 * ((b + (b + (-2.0 * (c * (a / b))))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-90:
		tmp = -c / b
	elif b <= 1.5e+142:
		tmp = -0.5 * ((b + math.sqrt(((a * (c * -4.0)) + (b * b)))) / a)
	else:
		tmp = -0.5 * ((b + (b + (-2.0 * (c * (a / b))))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-90)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.5e+142)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(a * Float64(c * -4.0)) + Float64(b * b)))) / a));
	else
		tmp = Float64(-0.5 * Float64(Float64(b + Float64(b + Float64(-2.0 * Float64(c * Float64(a / b))))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e-90)
		tmp = -c / b;
	elseif (b <= 1.5e+142)
		tmp = -0.5 * ((b + sqrt(((a * (c * -4.0)) + (b * b)))) / a);
	else
		tmp = -0.5 * ((b + (b + (-2.0 * (c * (a / b))))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-90], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.5e+142], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.60000000000000004e-90

   1. Initial program 17.1%

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

   2. Taylor expanded in b around -inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   4. Simplified 82.3%

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

   if -1.60000000000000004e-90 < b < 1.49999999999999987e142

   1. Initial program 77.9%

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

      1. /-rgt-identity 77.9%

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

      2. metadata-eval 77.9%

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

      3. associate-/l* 77.7%

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

      4. associate-/r/ 77.7%

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

      5. *-commutative 77.7%

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

      6. metadata-eval 77.7%

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

      7. metadata-eval 77.7%

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

      8. associate-*l/ 77.7%

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

      9. associate-/r/ 77.7%

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

      10. times-frac 77.9%

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

      11. *-commutative 77.9%

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

      12. times-frac 77.9%

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

      13. metadata-eval 77.9%

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

      14. associate-/r/ 77.9%

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

      15. *-commutative 77.9%

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

      16. div-sub 77.9%

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

   3. Simplified 77.9%

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

      1. fma-udef 77.9%

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

   5. Applied egg-rr 77.9%

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

   if 1.49999999999999987e142 < b

   1. Initial program 49.0%

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

      1. /-rgt-identity 49.0%

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

      2. metadata-eval 49.0%

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

      3. associate-/l* 49.0%

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

      4. associate-/r/ 49.0%

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

      5. *-commutative 49.0%

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

      6. metadata-eval 49.0%

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

      7. metadata-eval 49.0%

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

      8. associate-*l/ 49.0%

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

      9. associate-/r/ 49.0%

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

      10. times-frac 49.0%

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

      11. *-commutative 49.0%

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

      12. times-frac 49.0%

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

      13. metadata-eval 49.0%

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

      14. associate-/r/ 49.0%

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

      15. *-commutative 49.0%

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

      16. div-sub 49.0%

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

   3. Simplified 49.0%

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

      1. fma-udef 49.0%

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

   5. Applied egg-rr 49.0%

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

   6. Taylor expanded in b around inf 87.0%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}{a} \]
   7. Step-by-step derivation

      1. associate-*r/ 100.0%

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

   8. Simplified 100.0%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}{a}\\ \end{array} \]

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-104}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-125)
   (/ (- c) b)
   (if (<= b 1.12e-104)
     (* -0.5 (/ (+ b (sqrt (* c (* a -4.0)))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-125) {
		tmp = -c / b;
	} else if (b <= 1.12e-104) {
		tmp = -0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.8d-125)) then
        tmp = -c / b
    else if (b <= 1.12d-104) then
        tmp = (-0.5d0) * ((b + sqrt((c * (a * (-4.0d0))))) / a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-125) {
		tmp = -c / b;
	} else if (b <= 1.12e-104) {
		tmp = -0.5 * ((b + Math.sqrt((c * (a * -4.0)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-125:
		tmp = -c / b
	elif b <= 1.12e-104:
		tmp = -0.5 * ((b + math.sqrt((c * (a * -4.0)))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-125)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.12e-104)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-125)
		tmp = -c / b;
	elseif (b <= 1.12e-104)
		tmp = -0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-125], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.12e-104], N[(-0.5 * N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.8000000000000001e-125

   1. Initial program 20.0%

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

   2. Taylor expanded in b around -inf 80.2%

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

      1. associate-*r/ 80.2%

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

      2. neg-mul-1 80.2%

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

   4. Simplified 80.2%

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

   if -3.8000000000000001e-125 < b < 1.12e-104

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in a around inf 66.4%

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

      1. *-commutative 66.4%

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

      2. *-commutative 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*l* 66.4%

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

   6. Simplified 66.4%

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

   if 1.12e-104 < b

   1. Initial program 71.0%

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

   2. Taylor expanded in b around inf 89.0%

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

      1. mul-1-neg 89.0%

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

      2. unsub-neg 89.0%

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

   4. Simplified 89.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-104}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 68.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = -c / b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 29.9%

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

   2. Taylor expanded in b around -inf 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

   4. Simplified 66.1%

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

   if -9.999999999999969e-311 < b

   1. Initial program 71.2%

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

   2. Taylor expanded in b around inf 66.9%

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

      1. mul-1-neg 66.9%

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

      2. unsub-neg 66.9%

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

   4. Simplified 66.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 42.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+34}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -1.02e+34) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e+34) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.02d+34)) then
        tmp = c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e+34) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.02e+34:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e+34)
		tmp = Float64(c / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.02e+34)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e+34], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.02e34

   1. Initial program 15.8%

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

   2. Taylor expanded in b around -inf 67.6%

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

      1. associate-/l* 66.6%

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

   4. Simplified 66.6%

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

      1. add-cbrt-cube 50.1%

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

      2. pow3 50.0%

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

      3. times-frac 50.0%

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

      4. metadata-eval 50.0%

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

      5. mul-1-neg 50.0%

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

      6. div-inv 50.1%

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

      7. associate-/l* 50.1%

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

      8. clear-num 50.1%

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

   6. Applied egg-rr 50.1%

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

      1. pow1/3 41.5%

         \[\leadsto \color{blue}{{\left({\left(-\frac{c}{\frac{a}{\frac{a}{b}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

      2. distribute-neg-frac 41.5%

         \[\leadsto {\left({\color{blue}{\left(\frac{-c}{\frac{a}{\frac{a}{b}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      3. associate-/r/ 44.2%

         \[\leadsto {\left({\left(\frac{-c}{\color{blue}{\frac{a}{a} \cdot b}}\right)}^{3}\right)}^{0.3333333333333333} \]

      4. pow1 44.2%

         \[\leadsto {\left({\left(\frac{-c}{\frac{\color{blue}{{a}^{1}}}{a} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

      5. pow1 44.2%

         \[\leadsto {\left({\left(\frac{-c}{\frac{{a}^{1}}{\color{blue}{{a}^{1}}} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. pow-div 44.2%

         \[\leadsto {\left({\left(\frac{-c}{\color{blue}{{a}^{\left(1 - 1\right)}} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. metadata-eval 44.2%

         \[\leadsto {\left({\left(\frac{-c}{{a}^{\color{blue}{0}} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

      8. metadata-eval 44.2%

         \[\leadsto {\left({\left(\frac{-c}{\color{blue}{1} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

      9. *-un-lft-identity 44.2%

         \[\leadsto {\left({\left(\frac{-c}{\color{blue}{b}}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Applied egg-rr 44.2%

      \[\leadsto \color{blue}{{\left({\left(\frac{-c}{b}\right)}^{3}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 56.3%

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

      2. rem-cbrt-cube 91.1%

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

      3. add-sqr-sqrt 40.4%

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

      4. sqrt-unprod 45.9%

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

      5. sqr-neg 45.9%

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

      6. sqrt-unprod 19.3%

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

      7. add-sqr-sqrt 33.1%

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

      8. expm1-log1p-u 33.0%

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

      9. expm1-udef 33.7%

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

   10. Applied egg-rr 33.7%

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

       1. expm1-def 33.0%

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

       2. expm1-log1p 33.1%

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

   12. Simplified 33.1%

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

   if -1.02e34 < b

   1. Initial program 64.4%

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

   2. Taylor expanded in b around inf 49.0%

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

      1. associate-*r/ 49.0%

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

      2. mul-1-neg 49.0%

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

   4. Simplified 49.0%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+34}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6: 68.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 29.9%

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

   2. Taylor expanded in b around -inf 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

   4. Simplified 66.1%

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

   if -9.999999999999969e-311 < b

   1. Initial program 71.2%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. mul-1-neg 66.2%

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

   4. Simplified 66.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7: 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 52.6%

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

2. Taylor expanded in b around -inf 21.7%

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

   1. associate-/l* 23.7%

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

4. Simplified 23.7%

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

   1. add-cbrt-cube 16.9%

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

   2. pow3 16.9%

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

   3. times-frac 16.9%

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

   4. metadata-eval 16.9%

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

   5. mul-1-neg 16.9%

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

   6. div-inv 16.9%

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

   7. associate-/l* 16.9%

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

   8. clear-num 16.9%

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

6. Applied egg-rr 16.9%

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

   1. pow1/3 13.4%

      \[\leadsto \color{blue}{{\left({\left(-\frac{c}{\frac{a}{\frac{a}{b}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   2. distribute-neg-frac 13.4%

      \[\leadsto {\left({\color{blue}{\left(\frac{-c}{\frac{a}{\frac{a}{b}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

   3. associate-/r/ 14.1%

      \[\leadsto {\left({\left(\frac{-c}{\color{blue}{\frac{a}{a} \cdot b}}\right)}^{3}\right)}^{0.3333333333333333} \]

   4. pow1 14.1%

      \[\leadsto {\left({\left(\frac{-c}{\frac{\color{blue}{{a}^{1}}}{a} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

   5. pow1 14.1%

      \[\leadsto {\left({\left(\frac{-c}{\frac{{a}^{1}}{\color{blue}{{a}^{1}}} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

   6. pow-div 14.1%

      \[\leadsto {\left({\left(\frac{-c}{\color{blue}{{a}^{\left(1 - 1\right)}} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

   7. metadata-eval 14.1%

      \[\leadsto {\left({\left(\frac{-c}{{a}^{\color{blue}{0}} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. metadata-eval 14.1%

      \[\leadsto {\left({\left(\frac{-c}{\color{blue}{1} \cdot b}\right)}^{3}\right)}^{0.3333333333333333} \]

   9. *-un-lft-identity 14.1%

      \[\leadsto {\left({\left(\frac{-c}{\color{blue}{b}}\right)}^{3}\right)}^{0.3333333333333333} \]

8. Applied egg-rr 14.1%

   \[\leadsto \color{blue}{{\left({\left(\frac{-c}{b}\right)}^{3}\right)}^{0.3333333333333333}} \]
9. Step-by-step derivation

   1. unpow1/3 18.5%

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

   2. rem-cbrt-cube 30.8%

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

   3. add-sqr-sqrt 13.6%

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

   4. sqrt-unprod 15.1%

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

   5. sqr-neg 15.1%

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

   6. sqrt-unprod 5.8%

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

   7. add-sqr-sqrt 10.3%

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

   8. expm1-log1p-u 9.7%

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

   9. expm1-udef 9.9%

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

10. Applied egg-rr 9.9%

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

    1. expm1-def 9.7%

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

    2. expm1-log1p 10.3%

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

12. Simplified 10.3%

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

13. Final simplification 10.3%

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

Developer target: 70.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023240 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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