Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.7%

Time: 21.9s

Alternatives: 8

Speedup: 29.0×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.7% 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: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(20 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
      (* -0.25 (* 20.0 (/ (* a (pow c 4.0)) (pow b 7.0))))))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-0.25 * (20.0 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (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 = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-0.25d0) * (20.0d0 * ((a * (c ** 4.0d0)) / (b ** 7.0d0)))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-0.25 * (20.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0)))))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-0.25 * (20.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0)))))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-0.25 * Float64(20.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0)))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-0.25 * (20.0 * ((a * (c ^ 4.0)) / (b ^ 7.0)))))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(20.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.3%

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

   1. *-commutative 16.3%

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

3. Simplified 16.3%

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

5. Taylor expanded in a around 0 97.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

6. Taylor expanded in c around 0 97.7%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\left(20 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)}\right)\right) \]

7. Final simplification 97.7%

   \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(20 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
8. Add Preprocessing

Alternative 2: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (- (* -2.0 (/ (* a (pow c 3.0)) (pow b 5.0))) (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 5.0))) - (pow(c, 2.0) / pow(b, 3.0)))) - (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 = (a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((-2.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((-2.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((-2.0 * ((a * (c ^ 3.0)) / (b ^ 5.0))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.3%

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

   1. *-commutative 16.3%

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

   2. +-commutative 16.3%

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

   3. sqr-neg 16.3%

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

   4. unsub-neg 16.3%

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

   5. sqr-neg 16.3%

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

   6. fma-neg 16.3%

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

   7. distribute-lft-neg-in 16.3%

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

   8. *-commutative 16.3%

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

   9. *-commutative 16.3%

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

   10. distribute-rgt-neg-in 16.3%

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

   11. metadata-eval 16.3%

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

3. Simplified 16.3%

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

5. Taylor expanded in c around inf 16.4%

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

6. Taylor expanded in a around 0 96.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
7. Step-by-step derivation

   1. +-commutative 96.7%

      \[\leadsto \color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right) + -1 \cdot \frac{c}{b}} \]

   2. mul-1-neg 96.7%

      \[\leadsto a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right) + \color{blue}{\left(-\frac{c}{b}\right)} \]

   3. unsub-neg 96.7%

      \[\leadsto \color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

   4. mul-1-neg 96.7%

      \[\leadsto a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \color{blue}{\left(-\frac{{c}^{2}}{{b}^{3}}\right)}\right) - \frac{c}{b} \]

   5. unsub-neg 96.7%

      \[\leadsto a \cdot \color{blue}{\left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

8. Simplified 96.7%

   \[\leadsto \color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

9. Final simplification 96.7%

   \[\leadsto a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 3: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / 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 * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * ((-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-2.0 * ((c * (a ^ 2.0)) / (b ^ 5.0))) - (a / (b ^ 3.0)))) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.3%

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

   1. *-commutative 16.3%

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

3. Simplified 16.3%

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

5. Taylor expanded in c around 0 96.4%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]

6. Final simplification 96.4%

   \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \]
7. Add Preprocessing

Alternative 4: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (fma a (pow (/ c b) 2.0) c) (- b)))

C

double code(double a, double b, double c) {
	return fma(a, pow((c / b), 2.0), c) / -b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b))
end

Wolfram

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

Derivation

1. Initial program 16.3%

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

   1. *-commutative 16.3%

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

   2. +-commutative 16.3%

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

   3. sqr-neg 16.3%

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

   4. unsub-neg 16.3%

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

   5. sqr-neg 16.3%

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

   6. fma-neg 16.3%

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

   7. distribute-lft-neg-in 16.3%

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

   8. *-commutative 16.3%

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

   9. *-commutative 16.3%

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

   10. distribute-rgt-neg-in 16.3%

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

   11. metadata-eval 16.3%

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

3. Simplified 16.3%

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

5. Taylor expanded in c around inf 16.4%

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

6. Taylor expanded in b around inf 94.5%

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

   1. distribute-lft-out 94.5%

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

   2. *-lft-identity 94.5%

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

   3. times-frac 94.5%

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

   4. metadata-eval 94.5%

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

   5. associate-/l* 94.5%

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

   6. unpow2 94.5%

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

   7. unpow2 94.5%

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

   8. unpow2 94.5%

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

   9. times-frac 94.5%

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

   10. swap-sqr 94.5%

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

   11. unpow1 94.5%

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

   12. pow-plus 94.5%

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

   13. metadata-eval 94.5%

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

8. Simplified 94.5%

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

9. Taylor expanded in b around inf 95.0%

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
10. Step-by-step derivation

    1. distribute-lft-out 95.0%

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

    2. associate-*r/ 95.0%

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

    3. mul-1-neg 95.0%

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

    4. distribute-neg-frac2 95.0%

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

    5. +-commutative 95.0%

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

    6. associate-/l* 95.0%

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

    7. fma-define 95.0%

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

    8. unpow2 95.0%

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

    9. unpow2 95.0%

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

    10. times-frac 95.0%

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

    11. unpow2 95.0%

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

11. Simplified 95.0%

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

12. Final simplification 95.0%

    \[\leadsto \frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b} \]
13. Add Preprocessing

Alternative 5: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* a (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / pow(b, 3.0))));
}

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 * (((-1.0d0) / b) - (a * (c / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - (a * (c / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(c / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - (a * (c / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.3%

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

   1. *-commutative 16.3%

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

3. Simplified 16.3%

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

5. Taylor expanded in c around 0 94.5%

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

   1. distribute-lft-out 94.5%

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

   2. associate-/l* 94.5%

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

7. Simplified 94.5%

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

8. Taylor expanded in c around 0 94.7%

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

   1. sub-neg 94.7%

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

   2. distribute-neg-frac 94.7%

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

   3. metadata-eval 94.7%

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

   4. +-commutative 94.7%

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

   5. mul-1-neg 94.7%

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

   6. unsub-neg 94.7%

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

   7. associate-/l* 94.7%

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

10. Simplified 94.7%

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

11. Final simplification 94.7%

    \[\leadsto c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \]
12. Add Preprocessing

Alternative 6: 95.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} \cdot a\\ \frac{-2 \cdot \frac{c \cdot a + t\_0 \cdot t\_0}{b}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ c b) a)))
   (/ (* -2.0 (/ (+ (* c a) (* t_0 t_0)) b)) (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) * a;
	return (-2.0 * (((c * a) + (t_0 * t_0)) / b)) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = (c / b) * a
    code = ((-2.0d0) * (((c * a) + (t_0 * t_0)) / b)) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) * a;
	return (-2.0 * (((c * a) + (t_0 * t_0)) / b)) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = (c / b) * a
	return (-2.0 * (((c * a) + (t_0 * t_0)) / b)) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) * a)
	return Float64(Float64(-2.0 * Float64(Float64(Float64(c * a) + Float64(t_0 * t_0)) / b)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (c / b) * a;
	tmp = (-2.0 * (((c * a) + (t_0 * t_0)) / b)) / (a * 2.0);
end

Wolfram

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

Derivation

1. Initial program 16.3%

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

   1. *-commutative 16.3%

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

   2. +-commutative 16.3%

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

   3. sqr-neg 16.3%

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

   4. unsub-neg 16.3%

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

   5. sqr-neg 16.3%

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

   6. fma-neg 16.3%

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

   7. distribute-lft-neg-in 16.3%

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

   8. *-commutative 16.3%

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

   9. *-commutative 16.3%

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

   10. distribute-rgt-neg-in 16.3%

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

   11. metadata-eval 16.3%

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

3. Simplified 16.3%

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

5. Taylor expanded in c around inf 16.4%

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

6. Taylor expanded in b around inf 94.5%

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

   1. distribute-lft-out 94.5%

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

   2. *-lft-identity 94.5%

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

   3. times-frac 94.5%

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

   4. metadata-eval 94.5%

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

   5. associate-/l* 94.5%

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

   6. unpow2 94.5%

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

   7. unpow2 94.5%

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

   8. unpow2 94.5%

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

   9. times-frac 94.5%

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

   10. swap-sqr 94.5%

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

   11. unpow1 94.5%

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

   12. pow-plus 94.5%

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

   13. metadata-eval 94.5%

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

8. Simplified 94.5%

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

   1. unpow2 94.5%

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

   2. *-commutative 94.5%

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

   3. *-commutative 94.5%

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

10. Applied egg-rr 94.5%

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

11. Final simplification 94.5%

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

Alternative 7: 90.5% accurate, 29.0× 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(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 16.3%

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

   1. *-commutative 16.3%

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

3. Simplified 16.3%

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

5. Taylor expanded in b around inf 91.2%

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

   1. associate-*r/ 91.2%

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

   2. mul-1-neg 91.2%

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

7. Simplified 91.2%

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

8. Final simplification 91.2%

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

Alternative 8: 3.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 16.3%

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

   1. *-commutative 16.3%

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

   2. +-commutative 16.3%

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

   3. sqr-neg 16.3%

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

   4. unsub-neg 16.3%

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

   5. sqr-neg 16.3%

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

   6. fma-neg 16.3%

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

   7. distribute-lft-neg-in 16.3%

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

   8. *-commutative 16.3%

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

   9. *-commutative 16.3%

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

   10. distribute-rgt-neg-in 16.3%

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

   11. metadata-eval 16.3%

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

3. Simplified 16.3%

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

5. Taylor expanded in c around inf 16.4%

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

   1. *-un-lft-identity 16.4%

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

   2. add-sqr-sqrt 16.9%

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

   3. prod-diff 18.0%

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

   4. add-sqr-sqrt 18.2%

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

   5. fma-neg 18.2%

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

   6. *-un-lft-identity 18.2%

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

   7. *-commutative 18.2%

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

   8. fma-define 18.2%

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

   9. add-sqr-sqrt 18.0%

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

7. Applied egg-rr 18.0%

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

8. Taylor expanded in c around 0 3.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 3.3%

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

   2. distribute-rgt1-in 3.3%

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

   3. metadata-eval 3.3%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.3%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.3%

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

10. Simplified 3.3%

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

11. Final simplification 3.3%

    \[\leadsto \frac{0}{a} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))