Numeric.AD.Rank1.Halley:findZero from ad-4.2.4

Average Error: 11.0 → 0.1

Time: 7.9s

Precision: binary64

Cost: 832

Math

\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

↓

\[x - \frac{-2}{\frac{t}{z} + \frac{-2}{\frac{y}{z}}} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))

↓

(FPCore (x y z t)
 :precision binary64
 (- x (/ -2.0 (+ (/ t z) (/ -2.0 (/ y z))))))

C

double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

↓

double code(double x, double y, double z, double t) {
	return x - (-2.0 / ((t / z) + (-2.0 / (y / z))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - ((-2.0d0) / ((t / z) + ((-2.0d0) / (y / z))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

↓

public static double code(double x, double y, double z, double t) {
	return x - (-2.0 / ((t / z) + (-2.0 / (y / z))));
}

Python

def code(x, y, z, t):
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))

↓

def code(x, y, z, t):
	return x - (-2.0 / ((t / z) + (-2.0 / (y / z))))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))))
end

↓

function code(x, y, z, t)
	return Float64(x - Float64(-2.0 / Float64(Float64(t / z) + Float64(-2.0 / Float64(y / z)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
end

↓

function tmp = code(x, y, z, t)
	tmp = x - (-2.0 / ((t / z) + (-2.0 / (y / z))));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(x - N[(-2.0 / N[(N[(t / z), $MachinePrecision] + N[(-2.0 / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	11.0
Target	0.1
Herbie	0.1

\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \]

Derivation

1. Initial program 11.0

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

2. Simplified 2.9

   \[\leadsto \color{blue}{x - \frac{2}{\frac{\frac{z}{0.5} - \frac{y \cdot t}{z}}{y}}} \]
   Proof
   (-.f64 x (/.f64 2 (/.f64 (-.f64 (/.f64 z 1/2) (/.f64 (*.f64 y t) z)) y))): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 2 (/.f64 (-.f64 (/.f64 z (Rewrite<= metadata-eval (/.f64 1 2))) (/.f64 (*.f64 y t) z)) y))): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 2 (/.f64 (-.f64 (/.f64 z (/.f64 (Rewrite<= metadata-eval (/.f64 2 2)) 2)) (/.f64 (*.f64 y t) z)) y))): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 2 (/.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z 2) (/.f64 2 2))) (/.f64 (*.f64 y t) z)) y))): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 2 (/.f64 (-.f64 (/.f64 (*.f64 z 2) (Rewrite=> metadata-eval 1)) (/.f64 (*.f64 y t) z)) y))): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 2 (/.f64 (-.f64 (/.f64 (*.f64 z 2) (Rewrite<= *-inverses_binary64 (/.f64 z z))) (/.f64 (*.f64 y t) z)) y))): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 2 (/.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 z 2) z) z)) (/.f64 (*.f64 y t) z)) y))): 26 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 2 (/.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)) z)) y))): 0 points increase in error, 1 points decrease in error
   (-.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 y) (/.f64 (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)) z)))): 2 points increase in error, 11 points decrease in error
   (-.f64 x (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y 2)) (/.f64 (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)) z))): 0 points increase in error, 0 points decrease in error
   (-.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t))))): 41 points increase in error, 5 points decrease in error

3. Taylor expanded in z around 0 0.1

   \[\leadsto x - \frac{2}{\color{blue}{-1 \cdot \frac{t}{z} + 2 \cdot \frac{z}{y}}} \]

4. Simplified 0.1

   \[\leadsto x - \frac{2}{\color{blue}{z \cdot \frac{2}{y} - \frac{t}{z}}} \]
   Proof
   (-.f64 (*.f64 z (/.f64 2 y)) (/.f64 t z)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z 2) y)) (/.f64 t z)): 14 points increase in error, 14 points decrease in error
   (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 z y) 2)) (/.f64 t z)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 z y))) (/.f64 t z)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 (/.f64 z y)) (neg.f64 (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2 (/.f64 z y)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 t z)) (*.f64 2 (/.f64 z y)))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 0.1

   \[\leadsto x - \color{blue}{\left(0 + \frac{2}{z \cdot \frac{2}{y} - \frac{t}{z}}\right)} \]

6. Simplified 0.1

   \[\leadsto x - \color{blue}{\frac{-2}{\frac{t}{z} + \frac{-2}{\frac{y}{z}}}} \]
   Proof
   (/.f64 -2 (+.f64 (/.f64 t z) (/.f64 -2 (/.f64 y z)))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (+.f64 (/.f64 t z) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) (/.f64 y z)))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (+.f64 (/.f64 t z) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 (/.f64 y z)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (+.f64 (/.f64 t z) (neg.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 2 y) z))))): 13 points increase in error, 12 points decrease in error
   (/.f64 -2 (+.f64 (/.f64 t z) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 z (/.f64 2 y)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 t z)))) (neg.f64 (*.f64 z (/.f64 2 y))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (+.f64 (neg.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 t) z))) (neg.f64 (*.f64 z (/.f64 2 y))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (/.f64 (neg.f64 t) z) (*.f64 z (/.f64 2 y)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 z (/.f64 2 y)) (/.f64 (neg.f64 t) z))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 z (/.f64 2 y))) (neg.f64 (/.f64 (neg.f64 t) z))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 z (/.f64 2 y)))) (neg.f64 (/.f64 (neg.f64 t) z)))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (+.f64 (-.f64 0 (*.f64 z (/.f64 2 y))) (neg.f64 (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.f64 t z)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (+.f64 (-.f64 0 (*.f64 z (/.f64 2 y))) (Rewrite=> remove-double-neg_binary64 (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 (*.f64 z (/.f64 2 y)) (/.f64 t z))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 (*.f64 z (/.f64 2 y)) (/.f64 t z))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 (*.f64 z (/.f64 2 y)) (/.f64 t z))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 -2 -1) (-.f64 (*.f64 z (/.f64 2 y)) (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> metadata-eval 2) (-.f64 (*.f64 z (/.f64 2 y)) (/.f64 t z))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 2 (-.f64 (*.f64 z (/.f64 2 y)) (/.f64 t z))))): 0 points increase in error, 0 points decrease in error

7. Final simplification 0.1

   \[\leadsto x - \frac{-2}{\frac{t}{z} + \frac{-2}{\frac{y}{z}}} \]

Alternatives

Alternative 1
Error	0.1
Cost	832

\[x + \frac{-2}{\frac{\frac{z}{y}}{0.5} - \frac{t}{z}} \]

Alternative 2
Error	7.0
Cost	712

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -9 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-39}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	6.9
Cost	712

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-39}:\\ \;\;\;\;x - \frac{-2 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-298}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-194}:\\ \;\;\;\;z \cdot \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	11.5
Cost	584

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022332 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))