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

Average Error: 3.8 → 0.4

Time: 6.7s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -200000000:\\ \;\;\;\;t_1 + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{t}{y \cdot z} \cdot 0.3333333333333333\\ \end{array} \]

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
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= t -200000000.0)
     (+ t_1 (/ t (* (* z 3.0) y)))
     (if (<= t 1.55e-7)
       (+ (- x (/ (/ y z) 3.0)) (/ (/ (/ t y) z) 3.0))
       (+ t_1 (* (/ t (* y z)) 0.3333333333333333))))))

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) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if (t <= -200000000.0) {
		tmp = t_1 + (t / ((z * 3.0) * y));
	} else if (t <= 1.55e-7) {
		tmp = (x - ((y / z) / 3.0)) + (((t / y) / z) / 3.0);
	} else {
		tmp = t_1 + ((t / (y * z)) * 0.3333333333333333);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - (y / (z * 3.0d0))
    if (t <= (-200000000.0d0)) then
        tmp = t_1 + (t / ((z * 3.0d0) * y))
    else if (t <= 1.55d-7) then
        tmp = (x - ((y / z) / 3.0d0)) + (((t / y) / z) / 3.0d0)
    else
        tmp = t_1 + ((t / (y * z)) * 0.3333333333333333d0)
    end if
    code = tmp
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) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if (t <= -200000000.0) {
		tmp = t_1 + (t / ((z * 3.0) * y));
	} else if (t <= 1.55e-7) {
		tmp = (x - ((y / z) / 3.0)) + (((t / y) / z) / 3.0);
	} else {
		tmp = t_1 + ((t / (y * z)) * 0.3333333333333333);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = x - (y / (z * 3.0))
	tmp = 0
	if t <= -200000000.0:
		tmp = t_1 + (t / ((z * 3.0) * y))
	elif t <= 1.55e-7:
		tmp = (x - ((y / z) / 3.0)) + (((t / y) / z) / 3.0)
	else:
		tmp = t_1 + ((t / (y * z)) * 0.3333333333333333)
	return tmp

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)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (t <= -200000000.0)
		tmp = Float64(t_1 + Float64(t / Float64(Float64(z * 3.0) * y)));
	elseif (t <= 1.55e-7)
		tmp = Float64(Float64(x - Float64(Float64(y / z) / 3.0)) + Float64(Float64(Float64(t / y) / z) / 3.0));
	else
		tmp = Float64(t_1 + Float64(Float64(t / Float64(y * z)) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / (z * 3.0));
	tmp = 0.0;
	if (t <= -200000000.0)
		tmp = t_1 + (t / ((z * 3.0) * y));
	elseif (t <= 1.55e-7)
		tmp = (x - ((y / z) / 3.0)) + (((t / y) / z) / 3.0);
	else
		tmp = t_1 + ((t / (y * z)) * 0.3333333333333333);
	end
	tmp_2 = tmp;
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_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -200000000.0], N[(t$95$1 + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.55e-7], N[(N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t / y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.8
Target	1.6
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if t < -2e8

   1. Initial program 0.6

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

   if -2e8 < t < 1.55e-7

   1. Initial program 6.1

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

   2. Simplified 0.2

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

      [Start] 6.1	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 6.1	\[ \left(x - \color{blue}{\frac{\frac{y}{z}}{3}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 1.1	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]
      rational.json-simplify-44 [=>] 0.2	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}} \]
      rational.json-simplify-46 [=>] 0.2	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{\frac{t}{y}}{z}}{3}} \]

   if 1.55e-7 < t

   1. Initial program 0.7

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

   2. Simplified 7.9

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

      [Start] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 2.3	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]
      rational.json-simplify-44 [=>] 7.9	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}} \]

   3. Applied egg-rr 0.7

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{t}{y \cdot z} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -200000000:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot z} \cdot 0.3333333333333333\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1224

\[\begin{array}{l} t_1 := \left(x - \frac{0.3333333333333333}{z} \cdot y\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;t \leq -6300000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-9}:\\ \;\;\;\;\left(x - \frac{y}{z} \cdot 0.3333333333333333\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.8
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+85}:\\ \;\;\;\;\left(x - \frac{0.3333333333333333}{z} \cdot y\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \leq 4:\\ \;\;\;\;\left(x - \frac{y}{z} \cdot 0.3333333333333333\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	1224

\[\begin{array}{l} \mathbf{if}\;t \leq -100000000:\\ \;\;\;\;\left(x - \frac{0.3333333333333333}{z} \cdot y\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;t \leq 10^{-9}:\\ \;\;\;\;\left(x - \frac{y}{z} \cdot 0.3333333333333333\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot z} \cdot 0.3333333333333333\\ \end{array} \]

Alternative 4
Error	0.4
Cost	1224

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -2000000:\\ \;\;\;\;t_1 + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(x - \frac{y}{z} \cdot 0.3333333333333333\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{t}{y \cdot z} \cdot 0.3333333333333333\\ \end{array} \]

Alternative 5
Error	3.8
Cost	960

\[\left(x - \frac{0.3333333333333333}{z} \cdot y\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

Reproduce

herbie shell --seed 2023075 
(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))))