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

Average Error: 3.3 → 0.4

Time: 17.8s

Precision: binary64

Cost: 7752

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 10^{+21}:\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

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-61)
   (+ (- x (/ y (* z 3.0))) (/ t (* 3.0 (* z y))))
   (if (<= (* z 3.0) 1e+21)
     (+ 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-61) {
		tmp = (x - (y / (z * 3.0))) + (t / (3.0 * (z * y)));
	} else if ((z * 3.0) <= 1e+21) {
		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-61)
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(3.0 * Float64(z * y))));
	elseif (Float64(z * 3.0) <= 1e+21)
		tmp = Float64(x + Float64(Float64(y - Float64(t / y)) / Float64(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-61], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(3.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * 3.0), $MachinePrecision], 1e+21], N[(x + N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(z * -3.0), $MachinePrecision]), $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	3.3
Target	1.5
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z 3) < -4.9999999999999999e-61

   1. Initial program 0.5

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

   2. Taylor expanded in t around 0 0.5

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

   3. Simplified 0.5

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

      [Start] 0.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + 0.3333333333333333 \cdot \frac{t}{y \cdot z} \]
      metadata-eval [<=] 0.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{1}{3}} \cdot \frac{t}{y \cdot z} \]
      times-frac [<=] 0.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{1 \cdot t}{3 \cdot \left(y \cdot z\right)}} \]
      *-lft-identity [=>] 0.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{t}}{3 \cdot \left(y \cdot z\right)} \]
      remove-double-neg [<=] 0.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{-\left(-t\right)}}{3 \cdot \left(y \cdot z\right)} \]
      remove-double-neg [=>] 0.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{t}}{3 \cdot \left(y \cdot z\right)} \]
      *-commutative [=>] 0.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot z\right) \cdot 3}} \]

   if -4.9999999999999999e-61 < (*.f64 z 3) < 1e21

   1. Initial program 10.2

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

   2. Simplified 0.3

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

      [Start] 10.2	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-+l- [=>] 10.2	\[ \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      sub-neg [=>] 10.2	\[ \color{blue}{x + \left(-\left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      neg-mul-1 [=>] 10.2	\[ x + \color{blue}{-1 \cdot \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      distribute-lft-out-- [<=] 10.2	\[ x + \color{blue}{\left(-1 \cdot \frac{y}{z \cdot 3} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      associate-*r/ [=>] 10.2	\[ x + \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*l/ [<=] 10.2	\[ x + \left(\color{blue}{\frac{-1}{z \cdot 3} \cdot y} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*r/ [=>] 10.2	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1 \cdot t}{\left(z \cdot 3\right) \cdot y}}\right) \]
      times-frac [=>] 0.3	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1}{z \cdot 3} \cdot \frac{t}{y}}\right) \]
      distribute-lft-out-- [=>] 0.3	\[ x + \color{blue}{\frac{-1}{z \cdot 3} \cdot \left(y - \frac{t}{y}\right)} \]
      *-commutative [=>] 0.3	\[ x + \frac{-1}{\color{blue}{3 \cdot z}} \cdot \left(y - \frac{t}{y}\right) \]
      associate-/r* [=>] 0.3	\[ x + \color{blue}{\frac{\frac{-1}{3}}{z}} \cdot \left(y - \frac{t}{y}\right) \]
      metadata-eval [=>] 0.3	\[ x + \frac{\color{blue}{-0.3333333333333333}}{z} \cdot \left(y - \frac{t}{y}\right) \]

   3. Applied egg-rr 0.3

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

   if 1e21 < (*.f64 z 3)

   1. Initial program 0.3

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

   2. Simplified 0.3

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

      [Start] 0.3	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      sub-neg [=>] 0.3	\[ \color{blue}{\left(x + \left(-\frac{y}{z \cdot 3}\right)\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      +-commutative [=>] 0.3	\[ \color{blue}{\left(\left(-\frac{y}{z \cdot 3}\right) + x\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      neg-mul-1 [=>] 0.3	\[ \left(\color{blue}{-1 \cdot \frac{y}{z \cdot 3}} + x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-*r/ [=>] 0.3	\[ \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} + x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      *-commutative [=>] 0.3	\[ \left(\frac{-1 \cdot y}{\color{blue}{3 \cdot z}} + x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      times-frac [=>] 0.3	\[ \left(\color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      fma-def [=>] 0.3	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{y}{z}, x\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      metadata-eval [=>] 0.3	\[ \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      *-commutative [=>] 0.3	\[ \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) + \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 10^{+21}:\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	29.8
Cost	1504

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ t_2 := \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ t_3 := \frac{-0.3333333333333333}{\frac{z}{y}}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-213}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-254}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-36}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	35.0
Cost	1244

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{-77}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-298}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 63000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1060000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 3
Error	29.0
Cost	1240

\[\begin{array}{l} t_1 := \frac{-0.3333333333333333}{\frac{z}{y}}\\ t_2 := 0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{if}\;x \leq -7200000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-201}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	18.3
Cost	1240

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ t_2 := x + -0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -5.3 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-295}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-247}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-206}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	18.3
Cost	1240

\[\begin{array}{l} t_1 := x + -0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9.4 \cdot 10^{-268}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-298}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-240}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-209}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	0.4
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+60} \lor \neg \left(t \leq 2 \cdot 10^{-36}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \end{array} \]

Alternative 7
Error	0.5
Cost	1224

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+81}:\\ \;\;\;\;t_1 + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 8
Error	2.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-137} \lor \neg \left(y \leq 4 \cdot 10^{-6}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 9
Error	2.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-119} \lor \neg \left(y \leq 1.5 \cdot 10^{-5}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{1}{y}}{3} \cdot \frac{t}{z}\\ \end{array} \]

Alternative 10
Error	2.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-166} \lor \neg \left(y \leq 1.5 \cdot 10^{-5}\right):\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{1}{y}}{3} \cdot \frac{t}{z}\\ \end{array} \]

Alternative 11
Error	2.4
Cost	968

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-119}:\\ \;\;\;\;x + t_1 \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 102000:\\ \;\;\;\;x + \frac{\frac{1}{y}}{3} \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{\frac{z}{t_1}}\\ \end{array} \]

Alternative 12
Error	2.5
Cost	968

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{-166}:\\ \;\;\;\;x + \frac{\frac{t_1}{z}}{-3}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{\frac{1}{y}}{3} \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{z \cdot -3}\\ \end{array} \]

Alternative 13
Error	11.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+74} \lor \neg \left(x \leq 2.9 \cdot 10^{-19}\right):\\ \;\;\;\;x + -0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y - \frac{t}{y}}{z}\\ \end{array} \]

Alternative 14
Error	8.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-59} \lor \neg \left(y \leq 280000\right):\\ \;\;\;\;x + -0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

Alternative 15
Error	5.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-17} \lor \neg \left(y \leq 114000\right):\\ \;\;\;\;x + -0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 16
Error	28.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-22}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	37.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022354 
(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))))