Cubic critical

Percentage Accurate: 52.3% → 85.4%

Time: 14.3s

Alternatives: 11

Speedup: 16.4×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \left(c \cdot -3\right)\right)}{3}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e+130)
   (/ (fma -2.0 (/ b a) (* (/ -0.5 b) (* c -3.0))) 3.0)
   (if (<= b 4e-101)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e+130) {
		tmp = fma(-2.0, (b / a), ((-0.5 / b) * (c * -3.0))) / 3.0;
	} else if (b <= 4e-101) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e+130)
		tmp = Float64(fma(-2.0, Float64(b / a), Float64(Float64(-0.5 / b) * Float64(c * -3.0))) / 3.0);
	elseif (b <= 4e-101)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e+130], N[(N[(-2.0 * N[(b / a), $MachinePrecision] + N[(N[(-0.5 / b), $MachinePrecision] * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 4e-101], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.4999999999999997e130

   1. Initial program 44.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. Applied egg-rr 44.7%

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

      1. div-inv 44.7%

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

      2. associate-/r* 44.9%

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

      3. frac-2neg 44.9%

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

      4. metadata-eval 44.9%

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

   5. Applied egg-rr 44.9%

      \[\leadsto \color{blue}{\frac{-\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}}{3}} \]
   6. Step-by-step derivation

      1. distribute-neg-frac 44.9%

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

      2. sub-neg 44.9%

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

      3. +-commutative 44.9%

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

      4. distribute-neg-in 44.9%

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

      5. remove-double-neg 44.9%

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

      6. sub-neg 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in b around -inf 0.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{b}{a} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}}}{3} \]
   9. Step-by-step derivation

      1. fma-def 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{a}, -0.5 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}\right)}}{3} \]

      2. associate-*r/ 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \color{blue}{\frac{-0.5 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}\right)}{3} \]

      3. *-rgt-identity 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{\color{blue}{b \cdot 1}}\right)}{3} \]

      4. times-frac 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \color{blue}{\frac{-0.5}{b} \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{1}}\right)}{3} \]

      5. *-commutative 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \frac{\color{blue}{{\left(\sqrt{-3}\right)}^{2} \cdot c}}{1}\right)}{3} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \frac{\color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)} \cdot c}{1}\right)}{3} \]

      7. rem-square-sqrt 97.8%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \frac{\color{blue}{-3} \cdot c}{1}\right)}{3} \]

      8. associate-*r/ 97.8%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \color{blue}{\left(-3 \cdot \frac{c}{1}\right)}\right)}{3} \]

      9. /-rgt-identity 97.8%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \left(-3 \cdot \color{blue}{c}\right)\right)}{3} \]

      10. *-commutative 97.8%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \color{blue}{\left(c \cdot -3\right)}\right)}{3} \]

   10. Simplified 97.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \left(c \cdot -3\right)\right)}}{3} \]

   if -5.4999999999999997e130 < b < 4.00000000000000021e-101

   1. Initial program 80.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   if 4.00000000000000021e-101 < b

   1. Initial program 14.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 83.5%

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

      1. associate-*r/ 83.5%

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

   4. Simplified 83.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \left(c \cdot -3\right)\right)}{3}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-113)
   (/ (/ (* b -2.0) a) 3.0)
   (if (<= b 2e-89)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-113) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else if (b <= 2e-89) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.05d-113)) then
        tmp = ((b * (-2.0d0)) / a) / 3.0d0
    else if (b <= 2d-89) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-113) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else if (b <= 2e-89) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-113:
		tmp = ((b * -2.0) / a) / 3.0
	elif b <= 2e-89:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-113)
		tmp = Float64(Float64(Float64(b * -2.0) / a) / 3.0);
	elseif (b <= 2e-89)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.05e-113)
		tmp = ((b * -2.0) / a) / 3.0;
	elseif (b <= 2e-89)
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-113], N[(N[(N[(b * -2.0), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 2e-89], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.05e-113

   1. Initial program 72.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. Applied egg-rr 57.9%

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

      1. div-inv 57.9%

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

      2. associate-/r* 58.0%

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

      3. frac-2neg 58.0%

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

      4. metadata-eval 58.0%

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

   5. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{\frac{-\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}}{3}} \]
   6. Step-by-step derivation

      1. distribute-neg-frac 58.0%

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

      2. sub-neg 58.0%

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

      3. +-commutative 58.0%

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

      4. distribute-neg-in 58.0%

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

      5. remove-double-neg 58.0%

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

      6. sub-neg 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in b around -inf 82.3%

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

      1. *-commutative 82.3%

         \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{a}}{3} \]

   10. Simplified 82.3%

       \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{a}}{3} \]

   if -2.05e-113 < b < 2.00000000000000008e-89

   1. Initial program 70.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around 0 68.8%

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

   if 2.00000000000000008e-89 < b

   1. Initial program 14.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 83.5%

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

      1. associate-*r/ 83.5%

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

   4. Simplified 83.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-112)
   (/ (/ (* b -2.0) a) 3.0)
   (if (<= b 8.2e-93)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-112) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else if (b <= 8.2e-93) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.35d-112)) then
        tmp = ((b * (-2.0d0)) / a) / 3.0d0
    else if (b <= 8.2d-93) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-112) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else if (b <= 8.2e-93) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-112:
		tmp = ((b * -2.0) / a) / 3.0
	elif b <= 8.2e-93:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-112)
		tmp = Float64(Float64(Float64(b * -2.0) / a) / 3.0);
	elseif (b <= 8.2e-93)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.35e-112)
		tmp = ((b * -2.0) / a) / 3.0;
	elseif (b <= 8.2e-93)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-112], N[(N[(N[(b * -2.0), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 8.2e-93], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35e-112

   1. Initial program 72.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. Applied egg-rr 57.9%

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

      1. div-inv 57.9%

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

      2. associate-/r* 58.0%

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

      3. frac-2neg 58.0%

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

      4. metadata-eval 58.0%

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

   5. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{\frac{-\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}}{3}} \]
   6. Step-by-step derivation

      1. distribute-neg-frac 58.0%

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

      2. sub-neg 58.0%

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

      3. +-commutative 58.0%

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

      4. distribute-neg-in 58.0%

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

      5. remove-double-neg 58.0%

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

      6. sub-neg 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in b around -inf 82.3%

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

      1. *-commutative 82.3%

         \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{a}}{3} \]

   10. Simplified 82.3%

       \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{a}}{3} \]

   if -1.35e-112 < b < 8.1999999999999998e-93

   1. Initial program 70.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around 0 68.8%

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

      1. associate-*r* 68.8%

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

      2. *-commutative 68.8%

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

   4. Simplified 68.8%

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

   if 8.1999999999999998e-93 < b

   1. Initial program 14.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 83.5%

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

      1. associate-*r/ 83.5%

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

   4. Simplified 83.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \left(c \cdot -3\right)\right)}{3}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-113)
   (/ (fma -2.0 (/ b a) (* (/ -0.5 b) (* c -3.0))) 3.0)
   (if (<= b 1.9e-90)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-113) {
		tmp = fma(-2.0, (b / a), ((-0.5 / b) * (c * -3.0))) / 3.0;
	} else if (b <= 1.9e-90) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-113)
		tmp = Float64(fma(-2.0, Float64(b / a), Float64(Float64(-0.5 / b) * Float64(c * -3.0))) / 3.0);
	elseif (b <= 1.9e-90)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-113], N[(N[(-2.0 * N[(b / a), $MachinePrecision] + N[(N[(-0.5 / b), $MachinePrecision] * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 1.9e-90], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.0000000000000002e-113

   1. Initial program 72.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. Applied egg-rr 57.9%

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

      1. div-inv 57.9%

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

      2. associate-/r* 58.0%

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

      3. frac-2neg 58.0%

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

      4. metadata-eval 58.0%

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

   5. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{\frac{-\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}}{3}} \]
   6. Step-by-step derivation

      1. distribute-neg-frac 58.0%

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

      2. sub-neg 58.0%

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

      3. +-commutative 58.0%

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

      4. distribute-neg-in 58.0%

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

      5. remove-double-neg 58.0%

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

      6. sub-neg 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in b around -inf 0.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{b}{a} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}}}{3} \]
   9. Step-by-step derivation

      1. fma-def 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{a}, -0.5 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}\right)}}{3} \]

      2. associate-*r/ 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \color{blue}{\frac{-0.5 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}}\right)}{3} \]

      3. *-rgt-identity 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{\color{blue}{b \cdot 1}}\right)}{3} \]

      4. times-frac 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \color{blue}{\frac{-0.5}{b} \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{1}}\right)}{3} \]

      5. *-commutative 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \frac{\color{blue}{{\left(\sqrt{-3}\right)}^{2} \cdot c}}{1}\right)}{3} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \frac{\color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)} \cdot c}{1}\right)}{3} \]

      7. rem-square-sqrt 82.4%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \frac{\color{blue}{-3} \cdot c}{1}\right)}{3} \]

      8. associate-*r/ 82.4%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \color{blue}{\left(-3 \cdot \frac{c}{1}\right)}\right)}{3} \]

      9. /-rgt-identity 82.4%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \left(-3 \cdot \color{blue}{c}\right)\right)}{3} \]

      10. *-commutative 82.4%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \color{blue}{\left(c \cdot -3\right)}\right)}{3} \]

   10. Simplified 82.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \left(c \cdot -3\right)\right)}}{3} \]

   if -9.0000000000000002e-113 < b < 1.9e-90

   1. Initial program 70.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around 0 68.8%

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

      1. associate-*r* 68.8%

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

      2. *-commutative 68.8%

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

   4. Simplified 68.8%

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

   if 1.9e-90 < b

   1. Initial program 14.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 83.5%

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

      1. associate-*r/ 83.5%

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

   4. Simplified 83.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{-0.5}{b} \cdot \left(c \cdot -3\right)\right)}{3}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 5: 67.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310)
   (+ (* 0.5 (/ c b)) (* (/ b a) -0.6666666666666666))
   (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (0.5 * (c / b)) + ((b / a) * -0.6666666666666666);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (0.5d0 * (c / b)) + ((b / a) * (-0.6666666666666666d0))
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (0.5 * (c / b)) + ((b / a) * -0.6666666666666666);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (0.5 * (c / b)) + ((b / a) * -0.6666666666666666)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(0.5 * Float64(c / b)) + Float64(Float64(b / a) * -0.6666666666666666));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (0.5 * (c / b)) + ((b / a) * -0.6666666666666666);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 73.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 67.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -1.999999999999994e-310 < b

   1. Initial program 26.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 67.4%

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

      1. associate-*r/ 67.4%

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

   4. Simplified 67.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 6: 67.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (/ (* b -2.0) a) 3.0) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((b * (-2.0d0)) / a) / 3.0d0
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((b * -2.0) / a) / 3.0;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = ((b * -2.0) / a) / 3.0
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(Float64(b * -2.0) / a) / 3.0);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = ((b * -2.0) / a) / 3.0;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(N[(b * -2.0), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 73.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. Applied egg-rr 64.3%

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

      1. div-inv 64.3%

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

      2. associate-/r* 64.4%

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

      3. frac-2neg 64.4%

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

      4. metadata-eval 64.4%

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

   5. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{\frac{-\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}}{3}} \]
   6. Step-by-step derivation

      1. distribute-neg-frac 64.4%

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

      2. sub-neg 64.4%

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

      3. +-commutative 64.4%

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

      4. distribute-neg-in 64.4%

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

      5. remove-double-neg 64.4%

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

      6. sub-neg 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in b around -inf 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{a}}{3} \]

   10. Simplified 67.0%

       \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{a}}{3} \]

   if -1.999999999999994e-310 < b

   1. Initial program 26.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 67.4%

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

      1. associate-*r/ 67.4%

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

   4. Simplified 67.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 7: 43.8% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.1e+34) (* b (/ -0.6666666666666666 a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.1e+34) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.1d+34) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.5d0 * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.1e+34) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.1e+34:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.1e+34)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.1e+34)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.1e+34], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.09999999999999977e34

   1. Initial program 66.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. Applied egg-rr 60.1%

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

   4. Taylor expanded in b around -inf 48.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. associate-*r/ 48.1%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-*l/ 48.1%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   6. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   if 3.09999999999999977e34 < b

   1. Initial program 8.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 71.6%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. associate-/l* 74.1%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

   4. Simplified 74.1%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. frac-2neg 74.1%

         \[\leadsto \color{blue}{\frac{--1.5 \cdot \frac{a}{\frac{b}{c}}}{-3 \cdot a}} \]

      2. distribute-lft-neg-in 74.1%

         \[\leadsto \frac{--1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{\left(-3\right) \cdot a}} \]

      3. metadata-eval 74.1%

         \[\leadsto \frac{--1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{-3} \cdot a} \]

      4. *-commutative 74.1%

         \[\leadsto \frac{--1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{a \cdot -3}} \]

      5. distribute-frac-neg 74.1%

         \[\leadsto \color{blue}{-\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{a \cdot -3}} \]

      6. add-sqr-sqrt 35.3%

         \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{\sqrt{a \cdot -3} \cdot \sqrt{a \cdot -3}}} \]

      7. sqrt-unprod 38.7%

         \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{\sqrt{\left(a \cdot -3\right) \cdot \left(a \cdot -3\right)}}} \]

      8. swap-sqr 38.7%

         \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\color{blue}{\left(a \cdot a\right) \cdot \left(-3 \cdot -3\right)}}} \]

      9. metadata-eval 38.7%

         \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\left(a \cdot a\right) \cdot \color{blue}{9}}} \]

      10. metadata-eval 38.7%

          \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\left(a \cdot a\right) \cdot \color{blue}{\left(3 \cdot 3\right)}}} \]

      11. swap-sqr 38.7%

          \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\color{blue}{\left(a \cdot 3\right) \cdot \left(a \cdot 3\right)}}} \]

      12. *-commutative 38.7%

          \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\color{blue}{\left(3 \cdot a\right)} \cdot \left(a \cdot 3\right)}} \]

      13. *-commutative 38.7%

          \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\left(3 \cdot a\right) \cdot \color{blue}{\left(3 \cdot a\right)}}} \]

      14. sqrt-unprod 14.9%

          \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{\sqrt{3 \cdot a} \cdot \sqrt{3 \cdot a}}} \]

      15. add-sqr-sqrt 31.6%

          \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{3 \cdot a}} \]

      16. times-frac 31.6%

          \[\leadsto -\color{blue}{\frac{-1.5}{3} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}} \]

      17. metadata-eval 31.6%

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

      18. div-inv 31.6%

          \[\leadsto --0.5 \cdot \frac{\color{blue}{a \cdot \frac{1}{\frac{b}{c}}}}{a} \]

      19. clear-num 31.6%

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

   6. Applied egg-rr 31.6%

      \[\leadsto \color{blue}{--0.5 \cdot \frac{a \cdot \frac{c}{b}}{a}} \]
   7. Step-by-step derivation

      1. distribute-lft-neg-in 31.6%

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

      2. metadata-eval 31.6%

         \[\leadsto \color{blue}{0.5} \cdot \frac{a \cdot \frac{c}{b}}{a} \]

      3. associate-*r/ 31.5%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{a \cdot c}{b}}}{a} \]

      4. associate-*l/ 31.8%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{a}{b} \cdot c}}{a} \]

      5. *-commutative 31.8%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{c \cdot \frac{a}{b}}}{a} \]

   8. Simplified 31.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot \frac{a}{b}}{a}} \]

   9. Taylor expanded in c around 0 31.4%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 67.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-308}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.9e-308) (* b (/ -0.6666666666666666 a)) (* (/ -0.5 b) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.9e-308) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (-0.5 / b) * c;
	}
	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 <= 3.9d-308) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = ((-0.5d0) / b) * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.9e-308) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (-0.5 / b) * c;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.9e-308:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (-0.5 / b) * c
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.9e-308)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(-0.5 / b) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.9e-308)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (-0.5 / b) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.9e-308], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8999999999999999e-308

   1. Initial program 73.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. Applied egg-rr 64.3%

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

   4. Taylor expanded in b around -inf 67.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-*l/ 66.9%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   6. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   if 3.8999999999999999e-308 < b

   1. Initial program 26.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 53.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. associate-/l* 56.2%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

   4. Simplified 56.2%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. add-cbrt-cube 34.5%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right) \cdot \left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right) \cdot \left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)}}}{3 \cdot a} \]

      2. pow1/3 24.3%

         \[\leadsto \frac{\color{blue}{{\left(\left(\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right) \cdot \left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right) \cdot \left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}^{0.3333333333333333}}}{3 \cdot a} \]

      3. pow3 24.3%

         \[\leadsto \frac{{\color{blue}{\left({\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)}^{3}\right)}}^{0.3333333333333333}}{3 \cdot a} \]

      4. div-inv 24.3%

         \[\leadsto \frac{{\left({\left(-1.5 \cdot \color{blue}{\left(a \cdot \frac{1}{\frac{b}{c}}\right)}\right)}^{3}\right)}^{0.3333333333333333}}{3 \cdot a} \]

      5. clear-num 24.3%

         \[\leadsto \frac{{\left({\left(-1.5 \cdot \left(a \cdot \color{blue}{\frac{c}{b}}\right)\right)}^{3}\right)}^{0.3333333333333333}}{3 \cdot a} \]

   6. Applied egg-rr 24.3%

      \[\leadsto \frac{\color{blue}{{\left({\left(-1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

   7. Taylor expanded in a around -inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. *-rgt-identity 67.4%

         \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b \cdot 1}} \]

      3. times-frac 67.2%

         \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot \frac{c}{1}} \]

      4. /-rgt-identity 67.2%

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

   9. Simplified 67.2%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-308}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot c\\ \end{array} \]

Alternative 9: 67.6% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-309}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1e-309) (* (/ b a) -0.6666666666666666) (* (/ -0.5 b) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-309) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (-0.5 / b) * c;
	}
	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-309) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = ((-0.5d0) / b) * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-309) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (-0.5 / b) * c;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1e-309:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (-0.5 / b) * c
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1e-309)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(-0.5 / b) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1e-309)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (-0.5 / b) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1e-309], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.000000000000002e-309

   1. Initial program 73.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 67.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   4. Simplified 67.0%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if 1.000000000000002e-309 < b

   1. Initial program 26.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 53.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. associate-/l* 56.2%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

   4. Simplified 56.2%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. add-cbrt-cube 34.5%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right) \cdot \left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right) \cdot \left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)}}}{3 \cdot a} \]

      2. pow1/3 24.3%

         \[\leadsto \frac{\color{blue}{{\left(\left(\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right) \cdot \left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right) \cdot \left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}^{0.3333333333333333}}}{3 \cdot a} \]

      3. pow3 24.3%

         \[\leadsto \frac{{\color{blue}{\left({\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)}^{3}\right)}}^{0.3333333333333333}}{3 \cdot a} \]

      4. div-inv 24.3%

         \[\leadsto \frac{{\left({\left(-1.5 \cdot \color{blue}{\left(a \cdot \frac{1}{\frac{b}{c}}\right)}\right)}^{3}\right)}^{0.3333333333333333}}{3 \cdot a} \]

      5. clear-num 24.3%

         \[\leadsto \frac{{\left({\left(-1.5 \cdot \left(a \cdot \color{blue}{\frac{c}{b}}\right)\right)}^{3}\right)}^{0.3333333333333333}}{3 \cdot a} \]

   6. Applied egg-rr 24.3%

      \[\leadsto \frac{\color{blue}{{\left({\left(-1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

   7. Taylor expanded in a around -inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. *-rgt-identity 67.4%

         \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b \cdot 1}} \]

      3. times-frac 67.2%

         \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot \frac{c}{1}} \]

      4. /-rgt-identity 67.2%

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

   9. Simplified 67.2%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-309}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot c\\ \end{array} \]

Alternative 10: 67.7% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* (/ b a) -0.6666666666666666) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 73.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 67.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   4. Simplified 67.0%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if -1.999999999999994e-310 < b

   1. Initial program 26.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 67.4%

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

      1. associate-*r/ 67.4%

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

   4. Simplified 67.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 11: 11.0% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* 0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.5 * (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 = 0.5d0 * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return 0.5 * (c / b);
}

Python

def code(a, b, c):
	return 0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in b around inf 28.4%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
3. Step-by-step derivation

   1. associate-/l* 29.6%

      \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

4. Simplified 29.6%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. frac-2neg 29.6%

      \[\leadsto \color{blue}{\frac{--1.5 \cdot \frac{a}{\frac{b}{c}}}{-3 \cdot a}} \]

   2. distribute-lft-neg-in 29.6%

      \[\leadsto \frac{--1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{\left(-3\right) \cdot a}} \]

   3. metadata-eval 29.6%

      \[\leadsto \frac{--1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{-3} \cdot a} \]

   4. *-commutative 29.6%

      \[\leadsto \frac{--1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{a \cdot -3}} \]

   5. distribute-frac-neg 29.6%

      \[\leadsto \color{blue}{-\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{a \cdot -3}} \]

   6. add-sqr-sqrt 14.7%

      \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{\sqrt{a \cdot -3} \cdot \sqrt{a \cdot -3}}} \]

   7. sqrt-unprod 16.7%

      \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{\sqrt{\left(a \cdot -3\right) \cdot \left(a \cdot -3\right)}}} \]

   8. swap-sqr 16.7%

      \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\color{blue}{\left(a \cdot a\right) \cdot \left(-3 \cdot -3\right)}}} \]

   9. metadata-eval 16.7%

      \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\left(a \cdot a\right) \cdot \color{blue}{9}}} \]

   10. metadata-eval 16.7%

       \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\left(a \cdot a\right) \cdot \color{blue}{\left(3 \cdot 3\right)}}} \]

   11. swap-sqr 16.7%

       \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\color{blue}{\left(a \cdot 3\right) \cdot \left(a \cdot 3\right)}}} \]

   12. *-commutative 16.7%

       \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\color{blue}{\left(3 \cdot a\right)} \cdot \left(a \cdot 3\right)}} \]

   13. *-commutative 16.7%

       \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\sqrt{\left(3 \cdot a\right) \cdot \color{blue}{\left(3 \cdot a\right)}}} \]

   14. sqrt-unprod 5.5%

       \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{\sqrt{3 \cdot a} \cdot \sqrt{3 \cdot a}}} \]

   15. add-sqr-sqrt 11.7%

       \[\leadsto -\frac{-1.5 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{3 \cdot a}} \]

   16. times-frac 11.7%

       \[\leadsto -\color{blue}{\frac{-1.5}{3} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}} \]

   17. metadata-eval 11.7%

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

   18. div-inv 11.7%

       \[\leadsto --0.5 \cdot \frac{\color{blue}{a \cdot \frac{1}{\frac{b}{c}}}}{a} \]

   19. clear-num 11.7%

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

6. Applied egg-rr 11.7%

   \[\leadsto \color{blue}{--0.5 \cdot \frac{a \cdot \frac{c}{b}}{a}} \]
7. Step-by-step derivation

   1. distribute-lft-neg-in 11.7%

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

   2. metadata-eval 11.7%

      \[\leadsto \color{blue}{0.5} \cdot \frac{a \cdot \frac{c}{b}}{a} \]

   3. associate-*r/ 11.6%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{a \cdot c}{b}}}{a} \]

   4. associate-*l/ 11.7%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{a}{b} \cdot c}}{a} \]

   5. *-commutative 11.7%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{c \cdot \frac{a}{b}}}{a} \]

8. Simplified 11.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot \frac{a}{b}}{a}} \]

9. Taylor expanded in c around 0 11.6%

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

10. Final simplification 11.6%

    \[\leadsto 0.5 \cdot \frac{c}{b} \]

Reproduce

herbie shell --seed 2023307 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))