Quadratic roots, narrow range

Average Error: 28.4 → 0.5

Time: 16.6s

Precision: binary64

Cost: 7744

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (* (/ (* a (* c -4.0)) (+ b (sqrt (+ (* -4.0 (* a c)) (* b b))))) (/ 0.5 a)))

C

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

↓

double code(double a, double b, double c) {
	return ((a * (c * -4.0)) / (b + sqrt(((-4.0 * (a * c)) + (b * b))))) * (0.5 / 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

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((a * (c * (-4.0d0))) / (b + sqrt((((-4.0d0) * (a * c)) + (b * b))))) * (0.5d0 / 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);
}

↓

public static double code(double a, double b, double c) {
	return ((a * (c * -4.0)) / (b + Math.sqrt(((-4.0 * (a * c)) + (b * b))))) * (0.5 / a);
}

Python

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

↓

def code(a, b, c):
	return ((a * (c * -4.0)) / (b + math.sqrt(((-4.0 * (a * c)) + (b * b))))) * (0.5 / 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

↓

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(c * -4.0)) / Float64(b + sqrt(Float64(Float64(-4.0 * Float64(a * c)) + Float64(b * b))))) * Float64(0.5 / a))
end

MATLAB

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

↓

function tmp = code(a, b, c)
	tmp = ((a * (c * -4.0)) / (b + sqrt(((-4.0 * (a * c)) + (b * b))))) * (0.5 / 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]

↓

code[a_, b_, c_] := N[(N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.4

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

2. Simplified 28.4

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]
   Proof
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c 4))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (neg.f64 (Rewrite=> *-commutative_binary64 (*.f64 4 c))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 4) c)) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 (neg.f64 4) c)) (*.f64 b b)))) b) (/.f64 1/2 a)): 1 points increase in error, 1 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a (neg.f64 4)) c)) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (+.f64 (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4))) c) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (+.f64 (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 a))) c) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.f64 (*.f64 4 a)) c)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) (neg.f64 b))) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 2) a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 -1) (*.f64 2 a)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (neg.f64 -1)) (*.f64 2 a))): 22 points increase in error, 35 points decrease in error
   (Rewrite=> associate-/l*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (*.f64 2 a) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (*.f64 2 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite=> /-rgt-identity_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 30.0

   \[\leadsto \left(\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

4. Applied egg-rr 28.5

   \[\leadsto \left(\sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - 16 \cdot {\left(a \cdot c\right)}^{2}}{b \cdot b - a \cdot \left(c \cdot -4\right)}}} - b\right) \cdot \frac{0.5}{a} \]

5. Applied egg-rr 27.6

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

6. Simplified 0.5

   \[\leadsto \color{blue}{\frac{a \cdot \left(c \cdot -4\right) + 0}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}} \cdot \frac{0.5}{a} \]
   Proof
   (/.f64 (+.f64 (*.f64 a (*.f64 c -4)) 0) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (*.f64 a (*.f64 c -4)) (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 b b) (*.f64 b b)))) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 a (*.f64 c -4)) (*.f64 b b)) (*.f64 b b))) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))))): 148 points increase in error, 99 points decrease in error
   (/.f64 (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 a (*.f64 c -4)))) (*.f64 b b)) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= fma-udef_binary64 (fma.f64 b b (*.f64 a (*.f64 c -4)))) (*.f64 b b)) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))))): 21 points increase in error, 25 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (-.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) (*.f64 b b)) 1)) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (-.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) (*.f64 b b)) 1) (+.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 c -4)) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (-.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) (*.f64 b b)) 1) (+.f64 b (sqrt.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 a (*.f64 c -4))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (-.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) (*.f64 b b)) 1) (+.f64 b (sqrt.f64 (Rewrite<= fma-udef_binary64 (fma.f64 b b (*.f64 a (*.f64 c -4))))))): 4 points increase in error, 4 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) (*.f64 b b)) (/.f64 1 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -4)))))))): 28 points increase in error, 29 points decrease in error

7. Applied egg-rr 0.5

   \[\leadsto \frac{a \cdot \left(c \cdot -4\right) + 0}{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}}} \cdot \frac{0.5}{a} \]

8. Final simplification 0.5

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

Alternatives

Alternative 1
Error	28.4
Cost	7360

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

Alternative 2
Error	28.4
Cost	7360

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

Alternative 3
Error	55.6
Cost	576

\[\frac{\frac{b}{a}}{\frac{b}{a}} + -2 \]

Alternative 4
Error	56.7
Cost	448

\[-2 + b \cdot \frac{-0.5}{a} \]

Alternative 5
Error	56.7
Cost	448

\[b \cdot \frac{-0.5}{a} + -1 \]

Alternative 6
Error	56.7
Cost	384

\[\frac{a + b}{-a} \]

Alternative 7
Error	61.8
Cost	320

\[\frac{b + -2}{a} \]

Alternative 8
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2022334 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))