quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.7% → 84.7%

Time: 10.2s

Alternatives: 9

Speedup: 15.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c \cdot 0.5}{\sqrt[3]{b_2} \cdot \sqrt[3]{b_2 \cdot b_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.15e-32)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.2e+20)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) (* (cbrt b_2) (cbrt (* b_2 b_2))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e-32) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.2e+20) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / (cbrt(b_2) * cbrt((b_2 * b_2))));
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e-32) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.2e+20) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / (Math.cbrt(b_2) * Math.cbrt((b_2 * b_2))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.15e-32)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.2e+20)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / Float64(cbrt(b_2) * cbrt(Float64(b_2 * b_2)))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.15e-32], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.2e+20], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / N[(N[Power[b$95$2, 1/3], $MachinePrecision] * N[Power[N[(b$95$2 * b$95$2), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.14999999999999995e-32

   1. Initial program 12.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 87.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -2.14999999999999995e-32 < b_2 < 1.2e20

   1. Initial program 79.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 1.2e20 < b_2

   1. Initial program 63.4%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 97.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 97.3%

         \[\leadsto -2 \cdot \frac{b_2}{a} + \color{blue}{\frac{0.5 \cdot c}{b_2}} \]

      2. add-cube-cbrt 97.3%

         \[\leadsto -2 \cdot \frac{b_2}{a} + \frac{0.5 \cdot c}{\color{blue}{\left(\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}\right) \cdot \sqrt[3]{b_2}}} \]

      3. associate-/r* 97.3%

         \[\leadsto -2 \cdot \frac{b_2}{a} + \color{blue}{\frac{\frac{0.5 \cdot c}{\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}}}{\sqrt[3]{b_2}}} \]

      4. *-commutative 97.3%

         \[\leadsto -2 \cdot \frac{b_2}{a} + \frac{\frac{\color{blue}{c \cdot 0.5}}{\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}}}{\sqrt[3]{b_2}} \]

      5. cbrt-unprod 97.3%

         \[\leadsto -2 \cdot \frac{b_2}{a} + \frac{\frac{c \cdot 0.5}{\color{blue}{\sqrt[3]{b_2 \cdot b_2}}}}{\sqrt[3]{b_2}} \]

   4. Applied egg-rr 97.3%

      \[\leadsto -2 \cdot \frac{b_2}{a} + \color{blue}{\frac{\frac{c \cdot 0.5}{\sqrt[3]{b_2 \cdot b_2}}}{\sqrt[3]{b_2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ 97.3%

         \[\leadsto -2 \cdot \frac{b_2}{a} + \color{blue}{\frac{c \cdot 0.5}{\sqrt[3]{b_2} \cdot \sqrt[3]{b_2 \cdot b_2}}} \]

   6. Simplified 97.3%

      \[\leadsto -2 \cdot \frac{b_2}{a} + \color{blue}{\frac{c \cdot 0.5}{\sqrt[3]{b_2} \cdot \sqrt[3]{b_2 \cdot b_2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c \cdot 0.5}{\sqrt[3]{b_2} \cdot \sqrt[3]{b_2 \cdot b_2}}\\ \end{array} \]

Alternative 2: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{-32}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.26 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8e-32)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.26e+20)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-32) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.26e+20) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-8d-32)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.26d+20) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-32) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.26e+20) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8e-32:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.26e+20:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8e-32)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.26e+20)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8e-32)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.26e+20)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8e-32], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.26e+20], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.00000000000000045e-32

   1. Initial program 12.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 87.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -8.00000000000000045e-32 < b_2 < 1.26e20

   1. Initial program 79.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 1.26e20 < b_2

   1. Initial program 63.4%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 97.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{-32}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.26 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-32)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3.3e-37)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (/ (- (- b_2) (+ b_2 (* -0.5 (/ c (/ b_2 a))))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-32) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.3e-37) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.9d-32)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 3.3d-37) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = (-b_2 - (b_2 + ((-0.5d0) * (c / (b_2 / a))))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-32) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.3e-37) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.9e-32:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3.3e-37:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-32)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3.3e-37)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(Float64(-b_2) - Float64(b_2 + Float64(-0.5 * Float64(c / Float64(b_2 / a))))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.9e-32)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3.3e-37)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-32], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.3e-37], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[((-b$95$2) - N[(b$95$2 + N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.89999999999999996e-32

   1. Initial program 12.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 87.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -2.89999999999999996e-32 < b_2 < 3.29999999999999982e-37

   1. Initial program 76.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 69.6%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
   3. Step-by-step derivation

      1. mul-1-neg 69.6%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-c \cdot a}}}{a} \]

      2. distribute-rgt-neg-out 69.6%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   4. Simplified 69.6%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   if 3.29999999999999982e-37 < b_2

   1. Initial program 68.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 90.5%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(b_2 + -0.5 \cdot \frac{c \cdot a}{b_2}\right)}}{a} \]
   3. Step-by-step derivation

      1. associate-/l* 93.0%

         \[\leadsto \frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \color{blue}{\frac{c}{\frac{b_2}{a}}}\right)}{a} \]

   4. Simplified 93.0%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \]

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-31}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-31)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.15e-36)
     (/ (- (sqrt (* c (- a)))) a)
     (/ (- (- b_2) (+ b_2 (* -0.5 (/ c (/ b_2 a))))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-31) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.15e-36) {
		tmp = -sqrt((c * -a)) / a;
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7d-31)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.15d-36) then
        tmp = -sqrt((c * -a)) / a
    else
        tmp = (-b_2 - (b_2 + ((-0.5d0) * (c / (b_2 / a))))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-31) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.15e-36) {
		tmp = -Math.sqrt((c * -a)) / a;
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7e-31:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.15e-36:
		tmp = -math.sqrt((c * -a)) / a
	else:
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-31)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.15e-36)
		tmp = Float64(Float64(-sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(Float64(-b_2) - Float64(b_2 + Float64(-0.5 * Float64(c / Float64(b_2 / a))))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7e-31)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.15e-36)
		tmp = -sqrt((c * -a)) / a;
	else
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-31], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.15e-36], N[((-N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[((-b$95$2) - N[(b$95$2 + N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.99999999999999971e-31

   1. Initial program 12.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 87.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -6.99999999999999971e-31 < b_2 < 1.14999999999999998e-36

   1. Initial program 76.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 76.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 76.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fma-neg 76.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 76.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 76.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fma-neg 76.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 75.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. *-commutative 75.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. fma-udef 76.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. distribute-lft-neg-in 76.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. *-commutative 76.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. distribute-rgt-neg-in 76.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. fma-def 75.9%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. *-commutative 75.9%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

      15. fma-udef 76.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)}}{a} \]

      16. distribute-lft-neg-in 76.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)}}{a} \]

      17. *-commutative 76.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)}}{a} \]

      18. distribute-rgt-neg-in 76.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)}}{a} \]

      19. fma-def 75.9%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)}}{a} \]

   3. Applied egg-rr 75.9%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 75.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      2. count-2 75.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

      3. *-commutative 75.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   5. Simplified 75.9%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

   6. Taylor expanded in b_2 around 0 69.2%

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

      1. mul-1-neg 69.2%

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

      2. distribute-lft1-in 69.2%

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

      3. metadata-eval 69.2%

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

      4. mul0-lft 69.5%

         \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{0} - c \cdot a}}{a} \]

      5. metadata-eval 69.5%

         \[\leadsto \frac{-\sqrt{\color{blue}{0} - c \cdot a}}{a} \]

      6. neg-sub0 69.5%

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

      7. *-commutative 69.5%

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

      8. distribute-rgt-neg-in 69.5%

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

   8. Simplified 69.5%

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

   if 1.14999999999999998e-36 < b_2

   1. Initial program 68.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 90.5%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(b_2 + -0.5 \cdot \frac{c \cdot a}{b_2}\right)}}{a} \]
   3. Step-by-step derivation

      1. associate-/l* 93.0%

         \[\leadsto \frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \color{blue}{\frac{c}{\frac{b_2}{a}}}\right)}{a} \]

   4. Simplified 93.0%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-31}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \]

Alternative 5: 68.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* -0.5 c) b_2)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 31.7%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 66.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 66.4%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 74.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 68.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6: 44.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.4e-308) (/ 0.0 a) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e-308) {
		tmp = 0.0 / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.4d-308)) then
        tmp = 0.0d0 / a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e-308) {
		tmp = 0.0 / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.4e-308:
		tmp = 0.0 / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.4e-308)
		tmp = Float64(0.0 / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.4e-308)
		tmp = 0.0 / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.4e-308], N[(0.0 / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.40000000000000008e-308

   1. Initial program 31.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 30.1%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

      2. pow2 30.1%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 30.1%

         \[\leadsto \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 30.2%

         \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. metadata-eval 30.2%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 30.2%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around -inf 22.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot b_2 + b_2}}{a} \]
   5. Step-by-step derivation

      1. distribute-lft1-in 22.5%

         \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

      2. metadata-eval 22.5%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 22.5%

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. Simplified 22.5%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   if -2.40000000000000008e-308 < b_2

   1. Initial program 74.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 68.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 7: 68.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-308) (/ (* -0.5 c) b_2) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-308) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.2d-308)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-308) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-308:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-308)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.2e-308)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-308], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.2000000000000001e-308

   1. Initial program 31.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 66.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -3.2000000000000001e-308 < b_2

   1. Initial program 74.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 68.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 8: 23.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.4e-308) (/ 0.0 a) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e-308) {
		tmp = 0.0 / a;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.4d-308)) then
        tmp = 0.0d0 / a
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e-308) {
		tmp = 0.0 / a;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.4e-308:
		tmp = 0.0 / a
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.4e-308)
		tmp = Float64(0.0 / a);
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.4e-308)
		tmp = 0.0 / a;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.4e-308], N[(0.0 / a), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.40000000000000008e-308

   1. Initial program 31.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 30.1%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

      2. pow2 30.1%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 30.1%

         \[\leadsto \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 30.2%

         \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. metadata-eval 30.2%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 30.2%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around -inf 22.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot b_2 + b_2}}{a} \]
   5. Step-by-step derivation

      1. distribute-lft1-in 22.5%

         \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

      2. metadata-eval 22.5%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 22.5%

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. Simplified 22.5%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   if -2.40000000000000008e-308 < b_2

   1. Initial program 74.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 47.6%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
   3. Step-by-step derivation

      1. mul-1-neg 47.6%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-c \cdot a}}}{a} \]

      2. distribute-rgt-neg-out 47.6%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   4. Simplified 47.6%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   5. Taylor expanded in b_2 around inf 26.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 26.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

      2. mul-1-neg 26.8%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

   7. Simplified 26.8%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 9: 11.0% accurate, 37.3× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b_2, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0 / a;
}

Python

def code(a, b_2, c):
	return 0.0 / a

Julia

function code(a, b_2, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. add-sqr-sqrt 51.0%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

   2. pow2 51.0%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]

   3. pow1/2 51.0%

      \[\leadsto \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

   4. sqrt-pow1 51.1%

      \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

   5. metadata-eval 51.1%

      \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

3. Applied egg-rr 51.1%

   \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

4. Taylor expanded in b_2 around -inf 13.1%

   \[\leadsto \frac{\color{blue}{-1 \cdot b_2 + b_2}}{a} \]
5. Step-by-step derivation

   1. distribute-lft1-in 13.1%

      \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

   2. metadata-eval 13.1%

      \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

   3. mul0-lft 13.1%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

6. Simplified 13.1%

   \[\leadsto \frac{\color{blue}{0}}{a} \]

7. Final simplification 13.1%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023214 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))