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

Average Error: 12.0 → 0.1

Time: 7.1s

Precision: binary64

Cost: 832

Math

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

↓

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

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 (- (* 2.0 (/ z y)) (/ t 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 / ((2.0 * (z / y)) - (t / 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) / ((2.0d0 * (z / y)) - (t / 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 / ((2.0 * (z / y)) - (t / 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 / ((2.0 * (z / y)) - (t / 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(2.0 * Float64(z / y)) - Float64(t / 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 / ((2.0 * (z / y)) - (t / 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[(2.0 * N[(z / y), $MachinePrecision]), $MachinePrecision] - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	12.0
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 12.0

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

2. Simplified 0.1

   \[\leadsto \color{blue}{x + \frac{-2}{2 \cdot \frac{z}{y} - \frac{t}{z}}} \]
   Proof
   (+.f64 x (/.f64 -2 (-.f64 (*.f64 2 (/.f64 z y)) (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) (-.f64 (*.f64 2 (/.f64 z y)) (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 z) y)) (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 z 2)) y) (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (-.f64 (/.f64 (*.f64 z 2) y) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 t z)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (-.f64 (/.f64 (*.f64 z 2) y) (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) (/.f64 t z))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (-.f64 (/.f64 (*.f64 z 2) y) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y t) (*.f64 y z)))))): 26 points increase in error, 1 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (-.f64 (/.f64 (*.f64 z 2) y) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 y t) z) y))))): 3 points increase in error, 24 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 z 2) (/.f64 (*.f64 y t) z)) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (/.f64 (-.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (*.f64 z 2) 1)) (/.f64 (*.f64 y t) z)) y))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (neg.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 (neg.f64 2) (/.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 z 2) z) z)) (/.f64 (*.f64 y t) z)) y))): 14 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (neg.f64 2) (/.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)) z)) y))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 (/.f64 (/.f64 (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)) z) y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 y) (/.f64 (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)) z))))): 3 points increase in error, 7 points decrease in error
   (+.f64 x (neg.f64 (/.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 (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)))))): 26 points increase in error, 6 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 x (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t))))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	7.4
Cost	712

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -3.572029334444871 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.976811952034945 \cdot 10^{-37}:\\ \;\;\;\;x + 2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2347087234185213 \cdot 10^{-194}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 4.049613007017049 \cdot 10^{-287}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	15.5
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6866299450447934 \cdot 10^{-265}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8163766971552834 \cdot 10^{-279}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	15.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022298 
(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)))))