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

Average Error: 3.7 → 0.6

Time: 13.1s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+90}:\\ \;\;\;\;t_1 + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t \leq 10^{+28}:\\ \;\;\;\;t_1 + {\left(\frac{z \cdot 3}{\frac{t}{y}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y}{3} \cdot \frac{-1}{z}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \]

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 -1e+90)
     (+ t_1 (/ t (* 3.0 (* y z))))
     (if (<= t 1e+28)
       (+ t_1 (pow (/ (* z 3.0) (/ t y)) -1.0))
       (+ (+ x (* (/ y 3.0) (/ -1.0 z))) (/ t (* y (* z 3.0))))))))

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 <= -1e+90) {
		tmp = t_1 + (t / (3.0 * (y * z)));
	} else if (t <= 1e+28) {
		tmp = t_1 + pow(((z * 3.0) / (t / y)), -1.0);
	} else {
		tmp = (x + ((y / 3.0) * (-1.0 / z))) + (t / (y * (z * 3.0)));
	}
	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 <= (-1d+90)) then
        tmp = t_1 + (t / (3.0d0 * (y * z)))
    else if (t <= 1d+28) then
        tmp = t_1 + (((z * 3.0d0) / (t / y)) ** (-1.0d0))
    else
        tmp = (x + ((y / 3.0d0) * ((-1.0d0) / z))) + (t / (y * (z * 3.0d0)))
    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 <= -1e+90) {
		tmp = t_1 + (t / (3.0 * (y * z)));
	} else if (t <= 1e+28) {
		tmp = t_1 + Math.pow(((z * 3.0) / (t / y)), -1.0);
	} else {
		tmp = (x + ((y / 3.0) * (-1.0 / z))) + (t / (y * (z * 3.0)));
	}
	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 <= -1e+90:
		tmp = t_1 + (t / (3.0 * (y * z)))
	elif t <= 1e+28:
		tmp = t_1 + math.pow(((z * 3.0) / (t / y)), -1.0)
	else:
		tmp = (x + ((y / 3.0) * (-1.0 / z))) + (t / (y * (z * 3.0)))
	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 <= -1e+90)
		tmp = Float64(t_1 + Float64(t / Float64(3.0 * Float64(y * z))));
	elseif (t <= 1e+28)
		tmp = Float64(t_1 + (Float64(Float64(z * 3.0) / Float64(t / y)) ^ -1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(y / 3.0) * Float64(-1.0 / z))) + Float64(t / Float64(y * Float64(z * 3.0))));
	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 <= -1e+90)
		tmp = t_1 + (t / (3.0 * (y * z)));
	elseif (t <= 1e+28)
		tmp = t_1 + (((z * 3.0) / (t / y)) ^ -1.0);
	else
		tmp = (x + ((y / 3.0) * (-1.0 / z))) + (t / (y * (z * 3.0)));
	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, -1e+90], N[(t$95$1 + N[(t / N[(3.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+28], N[(t$95$1 + N[Power[N[(N[(z * 3.0), $MachinePrecision] / N[(t / y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(y / 3.0), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.7
Target	1.7
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if t < -9.99999999999999966e89

   1. Initial program 0.8

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

   2. Taylor expanded in z around 0 0.8

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

   if -9.99999999999999966e89 < t < 9.99999999999999958e27

   1. Initial program 5.2

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

   2. Applied egg-rr 0.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{{\left(\frac{z \cdot 3}{\frac{t}{y}}\right)}^{-1}} \]

   if 9.99999999999999958e27 < t

   1. Initial program 0.6

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

   2. Applied egg-rr 0.6

      \[\leadsto \left(x - \color{blue}{\frac{1}{z} \cdot \frac{y}{3}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+90}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t \leq 10^{+28}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + {\left(\frac{z \cdot 3}{\frac{t}{y}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y}{3} \cdot \frac{-1}{z}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1480

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + t \cdot \frac{\frac{0.3333333333333333}{z}}{y}\\ \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{-42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1480

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-34}:\\ \;\;\;\;t_1 + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{-42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + t \cdot \frac{\frac{0.3333333333333333}{z}}{y}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1352

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_1 + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t \leq 1000000000000:\\ \;\;\;\;t_1 + \frac{1}{z} \cdot \frac{t}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y}{3} \cdot \frac{-1}{z}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	1224

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\frac{z}{\frac{0.3333333333333333}{y}}}\\ \mathbf{if}\;t \leq -5.5382486200106945 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.180317697471911 \cdot 10^{-101}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	20.0
Cost	976

\[\begin{array}{l} t_1 := \frac{0.3333333333333333}{y} \cdot \frac{t}{z}\\ t_2 := x + \frac{\frac{y}{-3}}{z}\\ \mathbf{if}\;x \leq -4.496786108815995 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.495385702741754 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.5418023261861877 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.761332450892425 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	20.0
Cost	976

\[\begin{array}{l} t_1 := \frac{0.3333333333333333}{y} \cdot \frac{t}{z}\\ t_2 := x + \frac{\frac{y}{-3}}{z}\\ \mathbf{if}\;x \leq -4.496786108815995 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.495385702741754 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.5418023261861877 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.761332450892425 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 7
Error	20.0
Cost	976

\[\begin{array}{l} t_1 := x + \frac{\frac{y}{-3}}{z}\\ \mathbf{if}\;x \leq -4.496786108815995 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.495385702741754 \cdot 10^{-227}:\\ \;\;\;\;\frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{elif}\;x \leq 2.5418023261861877 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.761332450892425 \cdot 10^{-69}:\\ \;\;\;\;\frac{0.3333333333333333}{y} \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 8
Error	20.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -4.496786108815995 \cdot 10^{-297}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 2.495385702741754 \cdot 10^{-227}:\\ \;\;\;\;\frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{elif}\;x \leq 2.5418023261861877 \cdot 10^{-152}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;x \leq 2.761332450892425 \cdot 10^{-69}:\\ \;\;\;\;\frac{0.3333333333333333}{y} \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 9
Error	2.0
Cost	968

\[\begin{array}{l} t_1 := x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.4875528400595768 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	5.9
Cost	840

\[\begin{array}{l} t_1 := x + \frac{\frac{y}{-3}}{z}\\ \mathbf{if}\;y \leq -37942.52708946362:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.4875528400595768 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	28.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -9.684310988347473 \cdot 10^{+46}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2116390885163935 \cdot 10^{+57}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	18.5
Cost	448

\[x + \frac{y}{\frac{z}{-0.3333333333333333}} \]

Alternative 13
Error	18.5
Cost	448

\[x + \frac{\frac{y}{-3}}{z} \]

Alternative 14
Error	37.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022304 
(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))))