Quadratic roots, full range

Percentage Accurate: 52.3% → 85.5%

Time: 10.9s

Alternatives: 9

Speedup: 2.5×

• could not determine a ground truth

  1. a = 2.582851024527224e-268
  2. b = 1.0131273003032775e+188
  3. c = 8.416319522179615e-54

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e+154)
   (- (/ b a))
   (if (<= b 3.8e-89)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+154) {
		tmp = -(b / a);
	} else if (b <= 3.8e-89) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d+154)) then
        tmp = -(b / a)
    else if (b <= 3.8d-89) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+154) {
		tmp = -(b / a);
	} else if (b <= 3.8e-89) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e+154:
		tmp = -(b / a)
	elif b <= 3.8e-89:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e+154)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 3.8e-89)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e+154)
		tmp = -(b / a);
	elseif (b <= 3.8e-89)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e+154], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 3.8e-89], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -2.09999999999999994e154

   1. Initial program 36.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      6. neg-lowering-neg.f64 98.0

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

   5. Simplified 98.0%

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

   if -2.09999999999999994e154 < b < 3.8000000000000001e-89

   1. Initial program 82.0%

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

   if 3.8000000000000001e-89 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 86.2

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

   5. Simplified 86.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+131}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+131)
   (- (/ b a))
   (if (<= b 1.5e-88)
     (* (/ -0.5 a) (- b (sqrt (fma b b (* c (* a -4.0))))))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+131) {
		tmp = -(b / a);
	} else if (b <= 1.5e-88) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+131)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.5e-88)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e+131], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.5e-88], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -2.29999999999999992e131

   1. Initial program 43.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      6. neg-lowering-neg.f64 98.2

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

   5. Simplified 98.2%

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

   if -2.29999999999999992e131 < b < 1.5e-88

   1. Initial program 80.9%

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

   4. Applied egg-rr 80.7%

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

   if 1.5e-88 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 86.2

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

   5. Simplified 86.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+131}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e-73)
   (fma b (/ c (* b b)) (- (/ b a)))
   (if (<= b 3e-86) (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0)) (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e-73) {
		tmp = fma(b, (c / (b * b)), -(b / a));
	} else if (b <= 3e-86) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.7e-73)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	elseif (b <= 3e-86)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.7e-73], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 3e-86], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7000000000000001e-73

   1. Initial program 67.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot \frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \color{blue}{b \cdot \frac{c}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{c}{{b}^{2}}, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{c}{{b}^{2}}}, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. /-lowering-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-lowering-neg.f64 93.1

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

   5. Simplified 93.1%

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

   if -1.7000000000000001e-73 < b < 3.0000000000000001e-86

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 68.0

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

   5. Simplified 68.0%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 68.0

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

   7. Applied egg-rr 68.0%

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

   if 3.0000000000000001e-86 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 86.2

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

   5. Simplified 86.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-89}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-72)
   (fma b (/ c (* b b)) (- (/ b a)))
   (if (<= b 8.5e-89)
     (* (- (sqrt (* c (* a -4.0))) b) (/ 0.5 a))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-72) {
		tmp = fma(b, (c / (b * b)), -(b / a));
	} else if (b <= 8.5e-89) {
		tmp = (sqrt((c * (a * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-72)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	elseif (b <= 8.5e-89)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-72], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 8.5e-89], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -7.60000000000000004e-72

   1. Initial program 67.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot \frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \color{blue}{b \cdot \frac{c}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{c}{{b}^{2}}, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{c}{{b}^{2}}}, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. /-lowering-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-lowering-neg.f64 93.1

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

   5. Simplified 93.1%

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

   if -7.60000000000000004e-72 < b < 8.49999999999999937e-89

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 68.0

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

   5. Simplified 68.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 67.8

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

   7. Applied egg-rr 67.8%

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

   if 8.49999999999999937e-89 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 86.2

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

   5. Simplified 86.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-89}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 69.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot \frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \color{blue}{b \cdot \frac{c}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{c}{{b}^{2}}, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{c}{{b}^{2}}}, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. /-lowering-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-lowering-neg.f64 69.4

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

   5. Simplified 69.4%

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

      1. distribute-frac-neg2 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. inv-pow N/A

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

      15. div-inv N/A

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

      16. /-lowering-/.f64 N/A

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

      17. /-lowering-/.f64 70.7

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

   7. Applied egg-rr 70.7%

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

   if -9.999999999999969e-311 < b

   1. Initial program 23.6%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 71.7

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

   5. Simplified 71.7%

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

4. Final simplification 71.2%

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

Alternative 6: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-308}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.9e-308) (- (/ b a)) (- (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-308) {
		tmp = -(b / a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.9d-308) then
        tmp = -(b / a)
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-308) {
		tmp = -(b / a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.9e-308:
		tmp = -(b / a)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.9e-308)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.9e-308)
		tmp = -(b / a);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.9e-308], (-N[(b / a), $MachinePrecision]), (-N[(c / b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 2.9e-308

   1. Initial program 69.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      6. neg-lowering-neg.f64 70.4

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

   5. Simplified 70.4%

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

   if 2.9e-308 < b

   1. Initial program 23.6%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-lowering-neg.f64 71.7

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

   5. Simplified 71.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-308}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-7}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 6.6e-7) (- (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.6e-7) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6.6d-7) then
        tmp = -(b / a)
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.6e-7) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.6e-7:
		tmp = -(b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.6e-7)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.6e-7)
		tmp = -(b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.6e-7], (-N[(b / a), $MachinePrecision]), N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.6000000000000003e-7

   1. Initial program 66.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      6. neg-lowering-neg.f64 56.2

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

   5. Simplified 56.2%

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

   if 6.6000000000000003e-7 < b

   1. Initial program 12.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot \frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \color{blue}{b \cdot \frac{c}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{c}{{b}^{2}}, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{c}{{b}^{2}}}, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. /-lowering-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-lowering-neg.f64 2.3

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

   5. Simplified 2.3%

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

   6. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 28.5

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

   8. Simplified 28.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-7}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. distribute-neg-in N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(b \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

   8. distribute-rgt-neg-out N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot \frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \color{blue}{b \cdot \frac{c}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{c}{{b}^{2}}, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{c}{{b}^{2}}}, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

   12. unpow2 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. distribute-neg-frac2 N/A

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

   15. mul-1-neg N/A

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

   16. /-lowering-/.f64 N/A

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

   17. mul-1-neg N/A

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

   18. neg-lowering-neg.f64 35.4

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

5. Simplified 35.4%

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

6. Taylor expanded in b around 0

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

   1. /-lowering-/.f64 12.5

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

8. Simplified 12.5%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
9. Add Preprocessing

Alternative 9: 2.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. distribute-neg-in N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(b \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

   8. distribute-rgt-neg-out N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot \frac{c}{{b}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \color{blue}{b \cdot \frac{c}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{c}{{b}^{2}}, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{c}{{b}^{2}}}, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

   12. unpow2 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. distribute-neg-frac2 N/A

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

   15. mul-1-neg N/A

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

   16. /-lowering-/.f64 N/A

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

   17. mul-1-neg N/A

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

   18. neg-lowering-neg.f64 35.4

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

5. Simplified 35.4%

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

   1. distribute-frac-neg2 N/A

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

   2. unsub-neg N/A

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

   3. associate-*r/ N/A

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

   4. frac-2neg N/A

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

   5. frac-sub N/A

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

   6. sqr-neg N/A

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

   7. unpow3 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. pow-pow N/A

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

   11. pow2 N/A

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

   12. sqr-neg N/A

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

   13. pow2 N/A

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

   14. pow-pow N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. unpow3 N/A

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

   18. frac-sub N/A

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

   19. associate-*r/ N/A

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

7. Applied egg-rr 2.3%

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

8. Taylor expanded in c around 0

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

   1. /-lowering-/.f64 2.5

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

10. Simplified 2.5%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))