Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Average Error: 3.7 → 1.8

Time: 13.9s

Precision: binary64

Cost: 960

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

↓

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

↓

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y)))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

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 / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
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 - (y / (z * 3.0d0))) + ((t / (z * 3.0d0)) / y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

↓

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y)

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

↓

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(z * 3.0)) / y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

↓

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Target

Original	3.7
Target	1.8
Herbie	1.8

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation

1. Initial program 3.7

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

2. Simplified 1.8

   \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}} \]
   Proof

   [Start] 3.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   associate-/r* [=>] 1.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

3. Final simplification 1.8

   \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Alternatives

Alternative 1
Error	0.8
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-151} \lor \neg \left(t \leq 2.2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

Alternative 2
Error	32.0
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-145} \lor \neg \left(x \leq 1.35 \cdot 10^{-27}\right) \land x \leq 8000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	17.6
Cost	977

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-197}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-259} \lor \neg \left(y \leq 2 \cdot 10^{-129}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \end{array} \]

Alternative 4
Error	31.6
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 175000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	17.7
Cost	976

\[\begin{array}{l} t_1 := x + \frac{-0.3333333333333333}{\frac{z}{y}}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-197}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-259}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-129}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	17.7
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-259}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-126}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	17.7
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-154}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-199}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-259}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-127}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 8
Error	17.6
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-155}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-197}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-260}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.96 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{t}{3}}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 9
Error	1.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-95} \lor \neg \left(y \leq 1.4 \cdot 10^{-68}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 10
Error	1.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-96} \lor \neg \left(y \leq 10^{-68}\right):\\ \;\;\;\;x + \frac{-0.3333333333333333}{\frac{z}{y - \frac{t}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 11
Error	1.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-87} \lor \neg \left(y \leq 5.2 \cdot 10^{-68}\right):\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 12
Error	11.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-78}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{y - \frac{t}{y}}{z} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 13
Error	9.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-40}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-41}:\\ \;\;\;\;x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 14
Error	8.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-40}:\\ \;\;\;\;x + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 15
Error	6.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-40}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 16
Error	38.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))