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

Average Error: 4.98% → 0.99%

Time: 16.8s

Precision: binary64

Cost: 7752

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{3} \cdot \frac{1}{z}}{y}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

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) -5e+37)
   (+ (- x (/ y (* z 3.0))) (/ (* (/ t 3.0) (/ 1.0 z)) y))
   (if (<= (* z 3.0) 2e+18)
     (+ x (/ (/ (- y (/ t y)) z) -3.0))
     (+ (fma -0.3333333333333333 (/ y z) x) (/ 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) <= -5e+37) {
		tmp = (x - (y / (z * 3.0))) + (((t / 3.0) * (1.0 / z)) / y);
	} else if ((z * 3.0) <= 2e+18) {
		tmp = x + (((y - (t / y)) / z) / -3.0);
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x) + (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) <= -5e+37)
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(Float64(t / 3.0) * Float64(1.0 / z)) / y));
	elseif (Float64(z * 3.0) <= 2e+18)
		tmp = Float64(x + Float64(Float64(Float64(y - Float64(t / y)) / z) / -3.0));
	else
		tmp = Float64(fma(-0.3333333333333333, Float64(y / z), x) + Float64(t / Float64(Float64(z * 3.0) * y)));
	end
	return 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+37], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t / 3.0), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * 3.0), $MachinePrecision], 2e+18], N[(x + N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	4.98%
Target	2.51%
Herbie	0.99%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z 3) < -4.99999999999999989e37

   1. Initial program 0.63

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

   2. Simplified 1.79

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

      [Start] 0.63	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-/r* [=>] 1.79	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

   3. Applied egg-rr 1.73

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

   if -4.99999999999999989e37 < (*.f64 z 3) < 2e18

   1. Initial program 11.71

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

   2. Simplified 0.66

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

      [Start] 11.71	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-+l- [=>] 11.71	\[ \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      sub-neg [=>] 11.71	\[ \color{blue}{x + \left(-\left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      neg-mul-1 [=>] 11.71	\[ x + \color{blue}{-1 \cdot \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      distribute-lft-out-- [<=] 11.71	\[ x + \color{blue}{\left(-1 \cdot \frac{y}{z \cdot 3} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      associate-*r/ [=>] 11.71	\[ x + \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*l/ [<=] 11.78	\[ x + \left(\color{blue}{\frac{-1}{z \cdot 3} \cdot y} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*r/ [=>] 11.78	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1 \cdot t}{\left(z \cdot 3\right) \cdot y}}\right) \]
      times-frac [=>] 0.66	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1}{z \cdot 3} \cdot \frac{t}{y}}\right) \]
      distribute-lft-out-- [=>] 0.67	\[ x + \color{blue}{\frac{-1}{z \cdot 3} \cdot \left(y - \frac{t}{y}\right)} \]
      *-commutative [=>] 0.67	\[ x + \frac{-1}{\color{blue}{3 \cdot z}} \cdot \left(y - \frac{t}{y}\right) \]
      associate-/r* [=>] 0.66	\[ x + \color{blue}{\frac{\frac{-1}{3}}{z}} \cdot \left(y - \frac{t}{y}\right) \]
      metadata-eval [=>] 0.66	\[ x + \frac{\color{blue}{-0.3333333333333333}}{z} \cdot \left(y - \frac{t}{y}\right) \]

   3. Applied egg-rr 0.73

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

   if 2e18 < (*.f64 z 3)

   1. Initial program 0.61

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

   2. Simplified 0.62

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

      [Start] 0.61	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      sub-neg [=>] 0.61	\[ \color{blue}{\left(x + \left(-\frac{y}{z \cdot 3}\right)\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      +-commutative [=>] 0.61	\[ \color{blue}{\left(\left(-\frac{y}{z \cdot 3}\right) + x\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      neg-mul-1 [=>] 0.61	\[ \left(\color{blue}{-1 \cdot \frac{y}{z \cdot 3}} + x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-*r/ [=>] 0.61	\[ \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} + x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      *-commutative [=>] 0.61	\[ \left(\frac{-1 \cdot y}{\color{blue}{3 \cdot z}} + x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      times-frac [=>] 0.62	\[ \left(\color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      fma-def [=>] 0.62	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{y}{z}, x\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      metadata-eval [=>] 0.62	\[ \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      *-commutative [=>] 0.62	\[ \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) + \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.99

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{3} \cdot \frac{1}{z}}{y}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.65%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-45} \lor \neg \left(z \cdot 3 \leq 2 \cdot 10^{+18}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

Alternative 2
Error	0.99%
Cost	1480

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+37}:\\ \;\;\;\;t_1 + \frac{\frac{t}{z} \cdot 0.3333333333333333}{y}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

Alternative 3
Error	0.99%
Cost	1480

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+37}:\\ \;\;\;\;t_1 + \frac{\frac{t}{3} \cdot \frac{1}{z}}{y}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

Alternative 4
Error	3.17%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-45} \lor \neg \left(y \leq 4.2 \cdot 10^{-29}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{3}}{z}}{y}\\ \end{array} \]

Alternative 5
Error	3.18%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-47} \lor \neg \left(y \leq 7.6 \cdot 10^{-29}\right):\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{3}}{z}}{y}\\ \end{array} \]

Alternative 6
Error	50.33%
Cost	850

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+104} \lor \neg \left(y \leq -2.05 \cdot 10^{-18} \lor \neg \left(y \leq -4 \cdot 10^{-61}\right) \land y \leq 1.35 \cdot 10^{+103}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	50.16%
Cost	848

\[\begin{array}{l} t_1 := y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-61}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	50.17%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-61}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 9
Error	50.19%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{y}{\frac{z}{-0.3333333333333333}}\\ \mathbf{elif}\;y \leq -1.52 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-61}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 10
Error	50.16%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{\frac{z}{-0.3333333333333333}}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-61}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 11
Error	50.24%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{\frac{z}{-0.3333333333333333}}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-61}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 12
Error	18%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-10} \lor \neg \left(x \leq 760\right):\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{t}{y}}{z} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 13
Error	12.59%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;x + t \cdot \frac{\frac{0.3333333333333333}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 14
Error	12.09%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-34}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{0.3333333333333333}{\frac{z \cdot y}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 15
Error	12.03%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-32}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 16
Error	8.74%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-32}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{t \cdot \frac{0.3333333333333333}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 17
Error	8.75%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-37}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 18
Error	8.71%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-35}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 0.00024:\\ \;\;\;\;x + \frac{\frac{\frac{t}{3}}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 19
Error	8.76%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{\frac{-0.3333333333333333}{z}}{\frac{1}{y}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{3}}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 20
Error	27.59%
Cost	448

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

Alternative 21
Error	27.58%
Cost	448

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

Alternative 22
Error	27.51%
Cost	448

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

Alternative 23
Error	58.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023104 
(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))))