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

Average Error: 3.6 → 0.4

Time: 10.4s

Precision: binary64

Cost: 1480

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{0.3333333333333333 \cdot t}{y \cdot z}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\left(x - \frac{\frac{y}{3}}{z}\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\frac{0.3333333333333333}{z}}{y} \cdot t\\ \end{array} \]

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
 (if (<= (* z 3.0) -5e-24)
   (+ (- x (/ y (* z 3.0))) (/ (* 0.3333333333333333 t) (* y z)))
   (if (<= (* z 3.0) 2e-34)
     (+ (- x (/ (/ y 3.0) z)) (/ (/ (/ t y) z) 3.0))
     (+ (- x (/ (/ y z) 3.0)) (* (/ (/ 0.3333333333333333 z) y) t)))))

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 tmp;
	if ((z * 3.0) <= -5e-24) {
		tmp = (x - (y / (z * 3.0))) + ((0.3333333333333333 * t) / (y * z));
	} else if ((z * 3.0) <= 2e-34) {
		tmp = (x - ((y / 3.0) / z)) + (((t / y) / z) / 3.0);
	} else {
		tmp = (x - ((y / z) / 3.0)) + (((0.3333333333333333 / z) / y) * t);
	}
	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) :: tmp
    if ((z * 3.0d0) <= (-5d-24)) then
        tmp = (x - (y / (z * 3.0d0))) + ((0.3333333333333333d0 * t) / (y * z))
    else if ((z * 3.0d0) <= 2d-34) then
        tmp = (x - ((y / 3.0d0) / z)) + (((t / y) / z) / 3.0d0)
    else
        tmp = (x - ((y / z) / 3.0d0)) + (((0.3333333333333333d0 / z) / y) * t)
    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 tmp;
	if ((z * 3.0) <= -5e-24) {
		tmp = (x - (y / (z * 3.0))) + ((0.3333333333333333 * t) / (y * z));
	} else if ((z * 3.0) <= 2e-34) {
		tmp = (x - ((y / 3.0) / z)) + (((t / y) / z) / 3.0);
	} else {
		tmp = (x - ((y / z) / 3.0)) + (((0.3333333333333333 / z) / y) * t);
	}
	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):
	tmp = 0
	if (z * 3.0) <= -5e-24:
		tmp = (x - (y / (z * 3.0))) + ((0.3333333333333333 * t) / (y * z))
	elif (z * 3.0) <= 2e-34:
		tmp = (x - ((y / 3.0) / z)) + (((t / y) / z) / 3.0)
	else:
		tmp = (x - ((y / z) / 3.0)) + (((0.3333333333333333 / z) / y) * t)
	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)
	tmp = 0.0
	if (Float64(z * 3.0) <= -5e-24)
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(0.3333333333333333 * t) / Float64(y * z)));
	elseif (Float64(z * 3.0) <= 2e-34)
		tmp = Float64(Float64(x - Float64(Float64(y / 3.0) / z)) + Float64(Float64(Float64(t / y) / z) / 3.0));
	else
		tmp = Float64(Float64(x - Float64(Float64(y / z) / 3.0)) + Float64(Float64(Float64(0.3333333333333333 / z) / y) * t));
	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)
	tmp = 0.0;
	if ((z * 3.0) <= -5e-24)
		tmp = (x - (y / (z * 3.0))) + ((0.3333333333333333 * t) / (y * z));
	elseif ((z * 3.0) <= 2e-34)
		tmp = (x - ((y / 3.0) / z)) + (((t / y) / z) / 3.0);
	else
		tmp = (x - ((y / z) / 3.0)) + (((0.3333333333333333 / z) / y) * t);
	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_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -5e-24], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * t), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * 3.0), $MachinePrecision], 2e-34], N[(N[(x - N[(N[(y / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t / y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / z), $MachinePrecision] / y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Target

Original	3.6
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 (*.f64 z 3) < -4.9999999999999998e-24

   1. Initial program 0.5

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

   2. Simplified 5.8

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

      [Start] 0.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]
      rational.json-simplify-44 [=>] 5.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}} \]

   3. Applied egg-rr 5.8

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\left(\frac{t}{y} \cdot \frac{0.3333333333333333}{z} + 0\right)} \]

   4. Simplified 0.4

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

      [Start] 5.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \left(\frac{t}{y} \cdot \frac{0.3333333333333333}{z} + 0\right) \]
      rational.json-simplify-4 [=>] 5.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{t}{y} \cdot \frac{0.3333333333333333}{z}} \]
      rational.json-simplify-2 [<=] 5.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{0.3333333333333333}{z} \cdot \frac{t}{y}} \]
      rational.json-simplify-49 [<=] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{t \cdot \frac{0.3333333333333333}{z}}{y}} \]
      metadata-eval [<=] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t \cdot \frac{\color{blue}{\frac{1}{3}}}{z}}{y} \]
      rational.json-simplify-44 [<=] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t \cdot \color{blue}{\frac{\frac{1}{z}}{3}}}{y} \]
      rational.json-simplify-46 [<=] 0.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t \cdot \color{blue}{\frac{1}{z \cdot 3}}}{y} \]
      rational.json-simplify-49 [<=] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{1 \cdot t}{z \cdot 3}}}{y} \]
      rational.json-simplify-6 [=>] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{\color{blue}{t}}{z \cdot 3}}{y} \]
      rational.json-simplify-46 [=>] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{\frac{t}{z}}{3}}}{y} \]
      rational.json-simplify-44 [=>] 0.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{\frac{t}{3}}{z}}}{y} \]
      rational.json-simplify-44 [<=] 5.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{\frac{t}{3}}{y}}{z}} \]
      rational.json-simplify-47 [=>] 0.4	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{3}}{y \cdot z}} \]
      rational.json-simplify-6 [<=] 0.4	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{\color{blue}{1 \cdot t}}{3}}{y \cdot z} \]
      rational.json-simplify-49 [=>] 0.4	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{t \cdot \frac{1}{3}}}{y \cdot z} \]
      metadata-eval [=>] 0.4	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t \cdot \color{blue}{0.3333333333333333}}{y \cdot z} \]
      rational.json-simplify-2 [=>] 0.4	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{0.3333333333333333 \cdot t}}{y \cdot z} \]

   if -4.9999999999999998e-24 < (*.f64 z 3) < 1.99999999999999986e-34

   1. Initial program 11.7

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

   2. Simplified 0.5

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

      [Start] 11.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 11.9	\[ \left(x - \color{blue}{\frac{\frac{y}{z}}{3}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 3.2	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]
      rational.json-simplify-44 [=>] 0.4	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}} \]
      rational.json-simplify-46 [=>] 0.5	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{\frac{t}{y}}{z}}{3}} \]

   3. Applied egg-rr 0.4

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

   4. Applied egg-rr 0.4

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

   if 1.99999999999999986e-34 < (*.f64 z 3)

   1. Initial program 0.4

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

   2. Simplified 5.5

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

      [Start] 0.4	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 0.4	\[ \left(x - \color{blue}{\frac{\frac{y}{z}}{3}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 1.2	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]
      rational.json-simplify-44 [=>] 5.5	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}} \]
      rational.json-simplify-46 [=>] 5.5	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{\frac{t}{y}}{z}}{3}} \]

   3. Applied egg-rr 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{0.3333333333333333 \cdot t}{y \cdot z}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\left(x - \frac{\frac{y}{3}}{z}\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\frac{0.3333333333333333}{z}}{y} \cdot t\\ \end{array} \]

Alternatives

Alternative 1
Error	2.6
Cost	1224

\[\begin{array}{l} t_1 := \left(x - \frac{0.3333333333333333}{z} \cdot y\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-257}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.7
Cost	1224

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+246}:\\ \;\;\;\;\left(x - \frac{0.3333333333333333}{z} \cdot y\right) + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.9
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;x + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+41}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z} + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{y \cdot 3}\\ \end{array} \]

Alternative 4
Error	20.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -1.24 \cdot 10^{+117}:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t \leq 2000:\\ \;\;\;\;x + \frac{0.3333333333333333 \cdot \frac{t}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \end{array} \]

Alternative 5
Error	20.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -1.24 \cdot 10^{+117}:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+126}:\\ \;\;\;\;x + \frac{\frac{t}{y \cdot 3}}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{0.3333333333333333}{y}}{\frac{z}{t}}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	576

\[x + t \cdot \frac{0.3333333333333333}{y \cdot z} \]

Alternative 7
Error	20.3
Cost	576

\[x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y} \]

Reproduce

herbie shell --seed 2023064 
(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))))