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

Average Accuracy: 95.6% → 97.6%

Time: 17.3s

Precision: binary64

Cost: 1220

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 0.00018:\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

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) 0.00018)
   (+ x (/ (- y (/ t y)) (* z -3.0)))
   (+ (- 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) {
	double tmp;
	if ((z * 3.0) <= 0.00018) {
		tmp = x + ((y - (t / y)) / (z * -3.0));
	} else {
		tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
	}
	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) <= 0.00018d0) then
        tmp = x + ((y - (t / y)) / (z * (-3.0d0)))
    else
        tmp = (x - (y / (z * 3.0d0))) + ((t / (z * 3.0d0)) / y)
    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) <= 0.00018) {
		tmp = x + ((y - (t / y)) / (z * -3.0));
	} else {
		tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
	}
	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) <= 0.00018:
		tmp = x + ((y - (t / y)) / (z * -3.0))
	else:
		tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y)
	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) <= 0.00018)
		tmp = Float64(x + Float64(Float64(y - Float64(t / y)) / Float64(z * -3.0)));
	else
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(z * 3.0)) / y));
	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) <= 0.00018)
		tmp = x + ((y - (t / y)) / (z * -3.0));
	else
		tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
	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], 0.00018], N[(x + N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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	95.6%
Target	96.1%
Herbie	97.6%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 3) < 1.80000000000000011e-4

   1. Initial program 95.1%

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

   2. Simplified 98.7%

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

      [Start] 95.1	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-+l- [=>] 95.1	\[ \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      sub-neg [=>] 95.1	\[ \color{blue}{x + \left(-\left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      sub-neg [=>] 95.1	\[ x + \left(-\color{blue}{\left(\frac{y}{z \cdot 3} + \left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)}\right) \]
      distribute-neg-in [=>] 95.1	\[ x + \color{blue}{\left(\left(-\frac{y}{z \cdot 3}\right) + \left(-\left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)\right)} \]
      unsub-neg [=>] 95.1	\[ x + \color{blue}{\left(\left(-\frac{y}{z \cdot 3}\right) - \left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      neg-mul-1 [=>] 95.1	\[ x + \left(\color{blue}{-1 \cdot \frac{y}{z \cdot 3}} - \left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right) \]
      associate-*r/ [=>] 95.1	\[ x + \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} - \left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right) \]
      associate-*l/ [<=] 95.1	\[ x + \left(\color{blue}{\frac{-1}{z \cdot 3} \cdot y} - \left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right) \]
      distribute-neg-frac [=>] 95.1	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-t}{\left(z \cdot 3\right) \cdot y}}\right) \]
      neg-mul-1 [=>] 95.1	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \frac{\color{blue}{-1 \cdot t}}{\left(z \cdot 3\right) \cdot y}\right) \]
      times-frac [=>] 98.2	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1}{z \cdot 3} \cdot \frac{t}{y}}\right) \]
      distribute-lft-out-- [=>] 98.7	\[ x + \color{blue}{\frac{-1}{z \cdot 3} \cdot \left(y - \frac{t}{y}\right)} \]
      *-commutative [=>] 98.7	\[ x + \frac{-1}{\color{blue}{3 \cdot z}} \cdot \left(y - \frac{t}{y}\right) \]
      associate-/r* [=>] 98.7	\[ x + \color{blue}{\frac{\frac{-1}{3}}{z}} \cdot \left(y - \frac{t}{y}\right) \]
      metadata-eval [=>] 98.7	\[ x + \frac{\color{blue}{-0.3333333333333333}}{z} \cdot \left(y - \frac{t}{y}\right) \]

   3. Applied egg-rr 98.8%

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

      [Start] 98.7	\[ x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right) \]
      *-commutative [=>] 98.7	\[ x + \color{blue}{\left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}} \]
      clear-num [=>] 98.7	\[ x + \left(y - \frac{t}{y}\right) \cdot \color{blue}{\frac{1}{\frac{z}{-0.3333333333333333}}} \]
      un-div-inv [=>] 98.7	\[ x + \color{blue}{\frac{y - \frac{t}{y}}{\frac{z}{-0.3333333333333333}}} \]
      div-inv [=>] 98.8	\[ x + \frac{y - \frac{t}{y}}{\color{blue}{z \cdot \frac{1}{-0.3333333333333333}}} \]
      metadata-eval [=>] 98.8	\[ x + \frac{y - \frac{t}{y}}{z \cdot \color{blue}{-3}} \]

   if 1.80000000000000011e-4 < (*.f64 z 3)

   1. Initial program 98.6%

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

   2. Simplified 99.8%

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

      [Start] 98.6	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-/r* [=>] 99.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 0.00018:\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.9%
Cost	1220

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

Alternative 2
Accuracy	60.7%
Cost	1108

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ t_2 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-77}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	60.7%
Cost	1108

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{+25}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-38}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	62.4%
Cost	1108

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+27}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-80}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-38}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	76.9%
Cost	977

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-77} \lor \neg \left(y \leq 2.75 \cdot 10^{-64}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \end{array} \]

Alternative 6
Accuracy	77.1%
Cost	977

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;y \leq -4.25 \cdot 10^{-72} \lor \neg \left(y \leq 2.55 \cdot 10^{-64}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y}{\frac{t}{z}}}\\ \end{array} \]

Alternative 7
Accuracy	77.0%
Cost	977

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{1}{y} \cdot \frac{t}{z}\right)\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-78} \lor \neg \left(y \leq 7.4 \cdot 10^{-65}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y}{\frac{t}{z}}}\\ \end{array} \]

Alternative 8
Accuracy	88.6%
Cost	972

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-305}:\\ \;\;\;\;x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-38}:\\ \;\;\;\;x + \frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	96.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-304} \lor \neg \left(y \leq 7.5 \cdot 10^{-102}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \end{array} \]

Alternative 10
Accuracy	96.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-305} \lor \neg \left(y \leq 6.7 \cdot 10^{-102}\right):\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \end{array} \]

Alternative 11
Accuracy	80.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+26} \lor \neg \left(z \leq 1.25 \cdot 10^{+92}\right):\\ \;\;\;\;x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \end{array} \]

Alternative 12
Accuracy	88.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+27} \lor \neg \left(y \leq 2.2 \cdot 10^{-38}\right):\\ \;\;\;\;x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \end{array} \]

Alternative 13
Accuracy	48.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+81} \lor \neg \left(y \leq 1.05 \cdot 10^{+74}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Accuracy	48.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+81} \lor \neg \left(y \leq 1.02 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Accuracy	48.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+81}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 16
Accuracy	48.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{-0.3333333333333333}}\\ \end{array} \]

Alternative 17
Accuracy	30.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023160 
(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))))