quadm (p42, negative)

Percentage Accurate: 51.3% → 85.1%

Time: 10.7s

Alternatives: 8

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% 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: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-57}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 10^{+147}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-57)
   (/ (- c) b)
   (if (<= b 1e+147)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-57) {
		tmp = -c / b;
	} else if (b <= 1e+147) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} 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 <= (-1.15d-57)) then
        tmp = -c / b
    else if (b <= 1d+147) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    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 <= -1.15e-57) {
		tmp = -c / b;
	} else if (b <= 1e+147) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-57:
		tmp = -c / b
	elif b <= 1e+147:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-57)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1e+147)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-57)
		tmp = -c / b;
	elseif (b <= 1e+147)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-57], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1e+147], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15e-57

   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 91.1%

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

      1. associate-*r/ 91.1%

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

      2. neg-mul-1 91.1%

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

   4. Simplified 91.1%

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

   if -1.15e-57 < b < 9.9999999999999998e146

   1. Initial program 88.4%

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

   if 9.9999999999999998e146 < b

   1. Initial program 37.7%

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

   2. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-57}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 10^{+147}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 2: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-54}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+143}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e-54)
   (/ (- c) b)
   (if (<= b 2e+143)
     (* -0.5 (/ (+ b (sqrt (+ (* b b) (* a (* c -4.0))))) a))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-54) {
		tmp = -c / b;
	} else if (b <= 2e+143) {
		tmp = -0.5 * ((b + sqrt(((b * b) + (a * (c * -4.0))))) / a);
	} 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 <= (-2.45d-54)) then
        tmp = -c / b
    else if (b <= 2d+143) then
        tmp = (-0.5d0) * ((b + sqrt(((b * b) + (a * (c * (-4.0d0)))))) / a)
    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 <= -2.45e-54) {
		tmp = -c / b;
	} else if (b <= 2e+143) {
		tmp = -0.5 * ((b + Math.sqrt(((b * b) + (a * (c * -4.0))))) / a);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.45e-54:
		tmp = -c / b
	elif b <= 2e+143:
		tmp = -0.5 * ((b + math.sqrt(((b * b) + (a * (c * -4.0))))) / a)
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e-54)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2e+143)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))) / a));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.45e-54)
		tmp = -c / b;
	elseif (b <= 2e+143)
		tmp = -0.5 * ((b + sqrt(((b * b) + (a * (c * -4.0))))) / a);
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e-54], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2e+143], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4500000000000001e-54

   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 91.1%

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

      1. associate-*r/ 91.1%

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

      2. neg-mul-1 91.1%

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

   4. Simplified 91.1%

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

   if -2.4500000000000001e-54 < b < 2e143

   1. Initial program 88.4%

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

      1. /-rgt-identity 88.4%

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

      2. metadata-eval 88.4%

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

      3. associate-/l* 88.3%

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

      4. associate-/r/ 88.3%

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

      5. *-commutative 88.3%

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

      6. metadata-eval 88.3%

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

      7. metadata-eval 88.3%

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

      8. associate-*l/ 88.3%

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

      9. associate-/r/ 88.3%

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

      10. times-frac 88.4%

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

      11. *-commutative 88.4%

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

      12. times-frac 88.4%

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

      13. metadata-eval 88.4%

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

      14. associate-/r/ 88.4%

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

      15. *-commutative 88.4%

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

      16. div-sub 88.5%

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

   3. Simplified 87.7%

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

      1. fma-udef 87.7%

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

   5. Applied egg-rr 87.7%

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

   if 2e143 < b

   1. Initial program 37.7%

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

   2. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-54}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+143}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-57)
   (/ (- c) b)
   (if (<= b 2.7e-65)
     (* -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 <= -1.1e-57) {
		tmp = -c / b;
	} else if (b <= 2.7e-65) {
		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 <= (-1.1d-57)) then
        tmp = -c / b
    else if (b <= 2.7d-65) 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 <= -1.1e-57) {
		tmp = -c / b;
	} else if (b <= 2.7e-65) {
		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 <= -1.1e-57:
		tmp = -c / b
	elif b <= 2.7e-65:
		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 <= -1.1e-57)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2.7e-65)
		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 <= -1.1e-57)
		tmp = -c / b;
	elseif (b <= 2.7e-65)
		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, -1.1e-57], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2.7e-65], 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 < -1.09999999999999999e-57

   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 91.1%

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

      1. associate-*r/ 91.1%

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

      2. neg-mul-1 91.1%

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

   4. Simplified 91.1%

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

   if -1.09999999999999999e-57 < b < 2.6999999999999999e-65

   1. Initial program 82.1%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in a around inf 73.6%

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

      1. *-commutative 73.6%

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

      2. *-commutative 73.6%

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

      3. *-commutative 73.6%

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

      4. associate-*l* 73.6%

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

   6. Simplified 73.6%

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

   if 2.6999999999999999e-65 < b

   1. Initial program 70.4%

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

   2. Taylor expanded in b around inf 93.7%

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

      1. mul-1-neg 93.7%

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

      2. unsub-neg 93.7%

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

   4. Simplified 93.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 68.0% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-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 < -1.999999999999994e-310

   1. Initial program 34.9%

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

   2. Taylor expanded in b around -inf 64.9%

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

      1. associate-*r/ 64.9%

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

      2. neg-mul-1 64.9%

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

   4. Simplified 64.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 75.7%

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

   2. Taylor expanded in b around inf 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   4. Simplified 73.8%

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

4. Final simplification 69.4%

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

Alternative 5: 42.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -920000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -920000.0) (/ c b) (- (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -920000.0) {
		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 <= (-920000.0d0)) 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 <= -920000.0) {
		tmp = c / b;
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -920000.0:
		tmp = c / b
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -920000.0)
		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 <= -920000.0)
		tmp = c / b;
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -920000.0], N[(c / b), $MachinePrecision], (-N[(b / a), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < -9.2e5

   1. Initial program 16.3%

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

      1. clear-num 16.3%

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

      2. inv-pow 16.3%

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

   3. Applied egg-rr 7.6%

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

      1. unpow-1 7.6%

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

      2. unpow1/2 7.7%

         \[\leadsto \frac{1}{2 \cdot \frac{a}{b - \color{blue}{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}}}} \]

      3. metadata-eval 7.7%

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

      4. pow-sqr 7.7%

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

      5. pow-sqr 7.7%

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

      6. metadata-eval 7.7%

         \[\leadsto \frac{1}{2 \cdot \frac{a}{b - {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{\color{blue}{0.5}}}} \]

      7. unpow1/2 7.6%

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

      8. *-commutative 7.6%

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

      9. associate-*r* 7.6%

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

      10. remove-double-neg 7.6%

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

      11. fma-neg 7.6%

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

      12. *-commutative 7.6%

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

      13. distribute-lft-neg-in 7.6%

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

      14. distribute-rgt-neg-in 7.6%

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

      15. metadata-eval 7.6%

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

      16. associate-*r* 7.6%

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

      17. unsub-neg 7.6%

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

      18. fma-udef 7.6%

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

      19. distribute-rgt-neg-in 7.6%

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

   5. Simplified 7.6%

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

   6. Taylor expanded in a around 0 30.2%

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

   if -9.2e5 < b

   1. Initial program 70.7%

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

   2. Taylor expanded in b around inf 52.3%

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

      1. associate-*r/ 52.3%

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

      2. mul-1-neg 52.3%

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

   4. Simplified 52.3%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -920000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 6: 67.8% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 34.9%

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

   2. Taylor expanded in b around -inf 64.9%

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

      1. associate-*r/ 64.9%

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

      2. neg-mul-1 64.9%

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

   4. Simplified 64.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 75.7%

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

   2. Taylor expanded in b around inf 73.0%

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

      1. associate-*r/ 73.0%

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

      2. mul-1-neg 73.0%

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

   4. Simplified 73.0%

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

4. Final simplification 69.0%

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

Alternative 7: 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 55.8%

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

   1. clear-num 55.7%

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

   2. inv-pow 55.7%

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

3. Applied egg-rr 31.7%

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

   1. unpow-1 31.7%

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

   2. unpow1/2 31.8%

      \[\leadsto \frac{1}{2 \cdot \frac{a}{b - \color{blue}{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}}}} \]

   3. metadata-eval 31.8%

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

   4. pow-sqr 32.0%

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

   5. pow-sqr 31.8%

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

   6. metadata-eval 31.8%

      \[\leadsto \frac{1}{2 \cdot \frac{a}{b - {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{\color{blue}{0.5}}}} \]

   7. unpow1/2 31.7%

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

   8. *-commutative 31.7%

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

   9. associate-*r* 31.4%

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

   10. remove-double-neg 31.4%

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

   11. fma-neg 31.4%

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

   12. *-commutative 31.4%

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

   13. distribute-lft-neg-in 31.4%

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

   14. distribute-rgt-neg-in 31.4%

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

   15. metadata-eval 31.4%

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

   16. associate-*r* 31.7%

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

   17. unsub-neg 31.7%

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

   18. fma-udef 31.7%

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

   19. distribute-rgt-neg-in 31.7%

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

5. Simplified 31.7%

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

6. Taylor expanded in b around -inf 2.4%

   \[\leadsto \frac{1}{2 \cdot \color{blue}{\left(0.5 \cdot \frac{a}{b}\right)}} \]

7. Taylor expanded in a around 0 2.4%

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

8. Final simplification 2.4%

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

Alternative 8: 10.4% 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 55.8%

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

   1. clear-num 55.7%

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

   2. inv-pow 55.7%

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

3. Applied egg-rr 31.7%

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

   1. unpow-1 31.7%

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

   2. unpow1/2 31.8%

      \[\leadsto \frac{1}{2 \cdot \frac{a}{b - \color{blue}{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}}}} \]

   3. metadata-eval 31.8%

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

   4. pow-sqr 32.0%

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

   5. pow-sqr 31.8%

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

   6. metadata-eval 31.8%

      \[\leadsto \frac{1}{2 \cdot \frac{a}{b - {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{\color{blue}{0.5}}}} \]

   7. unpow1/2 31.7%

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

   8. *-commutative 31.7%

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

   9. associate-*r* 31.4%

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

   10. remove-double-neg 31.4%

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

   11. fma-neg 31.4%

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

   12. *-commutative 31.4%

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

   13. distribute-lft-neg-in 31.4%

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

   14. distribute-rgt-neg-in 31.4%

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

   15. metadata-eval 31.4%

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

   16. associate-*r* 31.7%

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

   17. unsub-neg 31.7%

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

   18. fma-udef 31.7%

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

   19. distribute-rgt-neg-in 31.7%

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

5. Simplified 31.7%

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

6. Taylor expanded in a around 0 10.5%

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

7. Final simplification 10.5%

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

Developer target: 69.5% 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 2023229 
(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)))