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

Average Error: 7.5 → 5.1

Time: 20.7s

Precision: binary64

Cost: 1736

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

↓

\[\begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ t_2 := 0.5 \cdot \frac{y \cdot x}{a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+276}:\\ \;\;\;\;t_2 + -4.5 \cdot \left(\frac{z}{a} \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 1.56 \cdot 10^{+133}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a} + \frac{x}{a} \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)) (t_2 (* 0.5 (/ (* y x) a))))
   (if (<= t_1 -2e+276)
     (+ t_2 (* -4.5 (* (/ z a) t)))
     (if (<= t_1 1.56e+133)
       (+ (* -4.5 (/ (* t z) a)) (* (/ x a) (* y 0.5)))
       (+ t_2 (* t (* (/ z a) -4.5)))))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double t_2 = 0.5 * ((y * x) / a);
	double tmp;
	if (t_1 <= -2e+276) {
		tmp = t_2 + (-4.5 * ((z / a) * t));
	} else if (t_1 <= 1.56e+133) {
		tmp = (-4.5 * ((t * z) / a)) + ((x / a) * (y * 0.5));
	} else {
		tmp = t_2 + (t * ((z / a) * -4.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    t_2 = 0.5d0 * ((y * x) / a)
    if (t_1 <= (-2d+276)) then
        tmp = t_2 + ((-4.5d0) * ((z / a) * t))
    else if (t_1 <= 1.56d+133) then
        tmp = ((-4.5d0) * ((t * z) / a)) + ((x / a) * (y * 0.5d0))
    else
        tmp = t_2 + (t * ((z / a) * (-4.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double t_2 = 0.5 * ((y * x) / a);
	double tmp;
	if (t_1 <= -2e+276) {
		tmp = t_2 + (-4.5 * ((z / a) * t));
	} else if (t_1 <= 1.56e+133) {
		tmp = (-4.5 * ((t * z) / a)) + ((x / a) * (y * 0.5));
	} else {
		tmp = t_2 + (t * ((z / a) * -4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

↓

def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	t_2 = 0.5 * ((y * x) / a)
	tmp = 0
	if t_1 <= -2e+276:
		tmp = t_2 + (-4.5 * ((z / a) * t))
	elif t_1 <= 1.56e+133:
		tmp = (-4.5 * ((t * z) / a)) + ((x / a) * (y * 0.5))
	else:
		tmp = t_2 + (t * ((z / a) * -4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	t_2 = Float64(0.5 * Float64(Float64(y * x) / a))
	tmp = 0.0
	if (t_1 <= -2e+276)
		tmp = Float64(t_2 + Float64(-4.5 * Float64(Float64(z / a) * t)));
	elseif (t_1 <= 1.56e+133)
		tmp = Float64(Float64(-4.5 * Float64(Float64(t * z) / a)) + Float64(Float64(x / a) * Float64(y * 0.5)));
	else
		tmp = Float64(t_2 + Float64(t * Float64(Float64(z / a) * -4.5)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	t_2 = 0.5 * ((y * x) / a);
	tmp = 0.0;
	if (t_1 <= -2e+276)
		tmp = t_2 + (-4.5 * ((z / a) * t));
	elseif (t_1 <= 1.56e+133)
		tmp = (-4.5 * ((t * z) / a)) + ((x / a) * (y * 0.5));
	else
		tmp = t_2 + (t * ((z / a) * -4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+276], N[(t$95$2 + N[(-4.5 * N[(N[(z / a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.56e+133], N[(N[(-4.5 * N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[(x / a), $MachinePrecision] * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	7.5
Target	5.3
Herbie	5.1

\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z 9) t) < -2.0000000000000001e276

   1. Initial program 47.2

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Simplified 46.8

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]
      Proof

      [Start] 47.2	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      rational.json-simplify-2 [=>] 47.2	\[ \frac{x \cdot y - \color{blue}{t \cdot \left(z \cdot 9\right)}}{a \cdot 2} \]
      rational.json-simplify-43 [=>] 46.8	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Applied egg-rr 63.6

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot y - z \cdot \left(9 \cdot t\right)} \cdot \left(\left(z \cdot \left(9 \cdot t\right) - x \cdot y\right) \cdot \left(z \cdot \left(9 \cdot t\right) - x \cdot y\right)\right)}}{a \cdot 2} \]

   4. Taylor expanded in t around 0 6.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a} + 0.5 \cdot \left(\left(-18 \cdot \frac{z}{a} - -9 \cdot \frac{z}{a}\right) \cdot t\right)} \]

   5. Simplified 6.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(\frac{z}{a} \cdot -4.5\right)} \]
      Proof

      [Start] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + 0.5 \cdot \left(\left(-18 \cdot \frac{z}{a} - -9 \cdot \frac{z}{a}\right) \cdot t\right) \]
      rational.json-simplify-43 [<=] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + \color{blue}{t \cdot \left(0.5 \cdot \left(-18 \cdot \frac{z}{a} - -9 \cdot \frac{z}{a}\right)\right)} \]
      rational.json-simplify-2 [=>] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(0.5 \cdot \left(\color{blue}{\frac{z}{a} \cdot -18} - -9 \cdot \frac{z}{a}\right)\right) \]
      rational.json-simplify-48 [=>] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(0.5 \cdot \color{blue}{\left(\frac{z}{a} \cdot \left(-18 - -9\right)\right)}\right) \]
      metadata-eval [=>] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(0.5 \cdot \left(\frac{z}{a} \cdot \color{blue}{-9}\right)\right) \]
      rational.json-simplify-43 [=>] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \color{blue}{\left(\frac{z}{a} \cdot \left(-9 \cdot 0.5\right)\right)} \]
      metadata-eval [=>] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(\frac{z}{a} \cdot \color{blue}{-4.5}\right) \]

   6. Applied egg-rr 6.1

      \[\leadsto 0.5 \cdot \frac{y \cdot x}{a} + \color{blue}{\left(t \cdot \left(\frac{z}{a} \cdot -4.5\right) + 0\right)} \]

   7. Simplified 6.1

      \[\leadsto 0.5 \cdot \frac{y \cdot x}{a} + \color{blue}{-4.5 \cdot \left(\frac{z}{a} \cdot t\right)} \]
      Proof

      [Start] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + \left(t \cdot \left(\frac{z}{a} \cdot -4.5\right) + 0\right) \]
      rational.json-simplify-4 [=>] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + \color{blue}{t \cdot \left(\frac{z}{a} \cdot -4.5\right)} \]
      rational.json-simplify-2 [=>] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \]
      rational.json-simplify-43 [=>] 6.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + \color{blue}{-4.5 \cdot \left(\frac{z}{a} \cdot t\right)} \]

   if -2.0000000000000001e276 < (*.f64 (*.f64 z 9) t) < 1.56e133

   1. Initial program 4.0

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Simplified 4.0

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]
      Proof

      [Start] 4.0	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      rational.json-simplify-2 [=>] 4.0	\[ \frac{x \cdot y - \color{blue}{t \cdot \left(z \cdot 9\right)}}{a \cdot 2} \]
      rational.json-simplify-43 [=>] 4.0	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Applied egg-rr 20.6

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot y - z \cdot \left(9 \cdot t\right)} \cdot \left(\left(z \cdot \left(9 \cdot t\right) - x \cdot y\right) \cdot \left(z \cdot \left(9 \cdot t\right) - x \cdot y\right)\right)}}{a \cdot 2} \]

   4. Taylor expanded in y around 0 4.9

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \left(y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right)\right)} \]

   5. Simplified 4.9

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right) \cdot \left(y \cdot 0.5\right)} \]
      Proof

      [Start] 4.9	\[ -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \left(y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right)\right) \]
      rational.json-simplify-2 [=>] 4.9	\[ -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(\left(2 \cdot \frac{x}{a} - \frac{x}{a}\right) \cdot y\right)} \]
      rational.json-simplify-43 [=>] 4.9	\[ -4.5 \cdot \frac{t \cdot z}{a} + \color{blue}{\left(2 \cdot \frac{x}{a} - \frac{x}{a}\right) \cdot \left(y \cdot 0.5\right)} \]

   6. Taylor expanded in x around 0 4.9

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + \color{blue}{\frac{x}{a}} \cdot \left(y \cdot 0.5\right) \]

   if 1.56e133 < (*.f64 (*.f64 z 9) t)

   1. Initial program 20.9

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Simplified 20.7

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]
      Proof

      [Start] 20.9	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      rational.json-simplify-2 [=>] 20.9	\[ \frac{x \cdot y - \color{blue}{t \cdot \left(z \cdot 9\right)}}{a \cdot 2} \]
      rational.json-simplify-43 [=>] 20.7	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Applied egg-rr 58.2

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot y - z \cdot \left(9 \cdot t\right)} \cdot \left(\left(z \cdot \left(9 \cdot t\right) - x \cdot y\right) \cdot \left(z \cdot \left(9 \cdot t\right) - x \cdot y\right)\right)}}{a \cdot 2} \]

   4. Taylor expanded in t around 0 7.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a} + 0.5 \cdot \left(\left(-18 \cdot \frac{z}{a} - -9 \cdot \frac{z}{a}\right) \cdot t\right)} \]

   5. Simplified 7.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(\frac{z}{a} \cdot -4.5\right)} \]
      Proof

      [Start] 7.1	\[ 0.5 \cdot \frac{y \cdot x}{a} + 0.5 \cdot \left(\left(-18 \cdot \frac{z}{a} - -9 \cdot \frac{z}{a}\right) \cdot t\right) \]
      rational.json-simplify-43 [<=] 7.0	\[ 0.5 \cdot \frac{y \cdot x}{a} + \color{blue}{t \cdot \left(0.5 \cdot \left(-18 \cdot \frac{z}{a} - -9 \cdot \frac{z}{a}\right)\right)} \]
      rational.json-simplify-2 [=>] 7.0	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(0.5 \cdot \left(\color{blue}{\frac{z}{a} \cdot -18} - -9 \cdot \frac{z}{a}\right)\right) \]
      rational.json-simplify-48 [=>] 7.0	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(0.5 \cdot \color{blue}{\left(\frac{z}{a} \cdot \left(-18 - -9\right)\right)}\right) \]
      metadata-eval [=>] 7.0	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(0.5 \cdot \left(\frac{z}{a} \cdot \color{blue}{-9}\right)\right) \]
      rational.json-simplify-43 [=>] 7.0	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \color{blue}{\left(\frac{z}{a} \cdot \left(-9 \cdot 0.5\right)\right)} \]
      metadata-eval [=>] 7.0	\[ 0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(\frac{z}{a} \cdot \color{blue}{-4.5}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+276}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} + -4.5 \cdot \left(\frac{z}{a} \cdot t\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 1.56 \cdot 10^{+133}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a} + \frac{x}{a} \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} + t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.1
Cost	1736

\[\begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ t_2 := 0.5 \cdot \frac{y \cdot x}{a} + -4.5 \cdot \left(\frac{z}{a} \cdot t\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1.56 \cdot 10^{+133}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a} + \frac{x}{a} \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	4.7
Cost	1480

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{t \cdot z}{a} + \frac{x}{a} \cdot \left(y \cdot 0.5\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	24.2
Cost	840

\[\begin{array}{l} t_1 := \frac{y \cdot x}{a \cdot 2}\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{-9 \cdot \left(t \cdot z\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	24.2
Cost	840

\[\begin{array}{l} t_1 := \frac{y \cdot x}{a \cdot 2}\\ \mathbf{if}\;x \leq -9.4 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	7.5
Cost	832

\[\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

Alternative 6
Error	24.2
Cost	712

\[\begin{array}{l} t_1 := \frac{y \cdot x}{a \cdot 2}\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-98}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	33.3
Cost	448

\[-4.5 \cdot \frac{t \cdot z}{a} \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))