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

Average Error: 4.0 → 1.3

Time: 12.2s

Precision: binary64

Cost: 3016

Math

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

↓

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ t_2 := t_1 + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+248}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{\frac{t}{y}}{z \cdot 3}\\ \end{array} \]

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)))) (t_2 (+ t_1 (/ t (* (* z 3.0) y)))))
   (if (<= t_2 (- INFINITY))
     (+ x (/ (/ (/ t y) z) 3.0))
     (if (<= t_2 4e+248) t_2 (+ t_1 (/ (/ 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 t_2 = t_1 + (t / ((z * 3.0) * y));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = x + (((t / y) / z) / 3.0);
	} else if (t_2 <= 4e+248) {
		tmp = t_2;
	} else {
		tmp = t_1 + ((t / y) / (z * 3.0));
	}
	return tmp;
}

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 t_2 = t_1 + (t / ((z * 3.0) * y));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = x + (((t / y) / z) / 3.0);
	} else if (t_2 <= 4e+248) {
		tmp = t_2;
	} else {
		tmp = t_1 + ((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))
	t_2 = t_1 + (t / ((z * 3.0) * y))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = x + (((t / y) / z) / 3.0)
	elif t_2 <= 4e+248:
		tmp = t_2
	else:
		tmp = t_1 + ((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)))
	t_2 = Float64(t_1 + Float64(t / Float64(Float64(z * 3.0) * y)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(x + Float64(Float64(Float64(t / y) / z) / 3.0));
	elseif (t_2 <= 4e+248)
		tmp = t_2;
	else
		tmp = Float64(t_1 + Float64(Float64(t / 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));
	t_2 = t_1 + (t / ((z * 3.0) * y));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = x + (((t / y) / z) / 3.0);
	elseif (t_2 <= 4e+248)
		tmp = t_2;
	else
		tmp = t_1 + ((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]}, Block[{t$95$2 = N[(t$95$1 + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(x + N[(N[(N[(t / y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+248], t$95$2, N[(t$95$1 + N[(N[(t / y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	4.0
Target	1.5
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.3

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

      [Start] 64.0	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 64.0	\[ \left(x - \color{blue}{\frac{\frac{y}{z}}{3}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 0.3	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]
      rational.json-simplify-44 [=>] 0.3	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}} \]
      rational.json-simplify-46 [=>] 0.3	\[ \left(x - \frac{\frac{y}{z}}{3}\right) + \color{blue}{\frac{\frac{\frac{t}{y}}{z}}{3}} \]

   3. Taylor expanded in x around inf 6.0

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

   if -inf.0 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < 4.00000000000000018e248

   1. Initial program 0.5

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

   if 4.00000000000000018e248 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))

   1. Initial program 15.0

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

   2. Simplified 5.6

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

      [Start] 15.0	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 2.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]
      rational.json-simplify-44 [=>] 5.6	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \leq -\infty:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{elif}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \leq 4 \cdot 10^{+248}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{y}}{z \cdot 3}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.7
Cost	3016

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y}}{z \cdot 3}\\ \end{array} \]

Alternative 2
Error	29.0
Cost	1108

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{z}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-246}:\\ \;\;\;\;\frac{-0.3333333333333333}{z} \cdot y\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	29.0
Cost	1108

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{if}\;x \leq -6 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{z}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-246}:\\ \;\;\;\;\frac{-0.3333333333333333}{z} \cdot y\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	28.9
Cost	1108

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-304}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{z}\\ \mathbf{elif}\;x \leq 10^{-283}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-246}:\\ \;\;\;\;\frac{-0.3333333333333333}{z} \cdot y\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	1.5
Cost	960

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

Alternative 6
Error	11.3
Cost	840

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	8.6
Cost	840

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -7.1 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	8.7
Cost	840

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	5.8
Cost	840

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	16.9
Cost	712

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-126}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	16.9
Cost	712

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-126}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \frac{t}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	15.8
Cost	712

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	28.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{-0.3333333333333333}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	28.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	28.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	37.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023069 
(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))))