Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, B

Average Error: 0.02% → 0%

Time: 7.1s

Precision: binary64

Cost: 13248

Math

\[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

↓

\[\mathsf{fma}\left(z, -0.5 \cdot y, \mathsf{fma}\left(0.125, x, t\right)\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

↓

(FPCore (x y z t) :precision binary64 (fma z (* -0.5 y) (fma 0.125 x t)))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

↓

double code(double x, double y, double z, double t) {
	return fma(z, (-0.5 * y), fma(0.125, x, t));
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

↓

function code(x, y, z, t)
	return fma(z, Float64(-0.5 * y), fma(0.125, x, t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(z * N[(-0.5 * y), $MachinePrecision] + N[(0.125 * x + t), $MachinePrecision]), $MachinePrecision]

Target

Original	0.02%
Target	0.01%
Herbie	0%

\[\left(\frac{x}{8} + t\right) - \frac{z}{2} \cdot y \]

Derivation

1. Initial program 0.02

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

2. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, -0.5 \cdot y, \mathsf{fma}\left(0.125, x, t\right)\right)} \]
   Proof

   [Start] 0.02	\[ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   sub-neg [=>] 0.02	\[ \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]
   +-commutative [=>] 0.02	\[ \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]
   associate-+l+ [=>] 0.02	\[ \color{blue}{\left(-\frac{y \cdot z}{2}\right) + \left(\frac{1}{8} \cdot x + t\right)} \]
   neg-mul-1 [=>] 0.02	\[ \color{blue}{-1 \cdot \frac{y \cdot z}{2}} + \left(\frac{1}{8} \cdot x + t\right) \]
   associate-*l/ [<=] 0.01	\[ -1 \cdot \color{blue}{\left(\frac{y}{2} \cdot z\right)} + \left(\frac{1}{8} \cdot x + t\right) \]
   associate-*r* [=>] 0.01	\[ \color{blue}{\left(-1 \cdot \frac{y}{2}\right) \cdot z} + \left(\frac{1}{8} \cdot x + t\right) \]
   *-commutative [=>] 0.01	\[ \color{blue}{z \cdot \left(-1 \cdot \frac{y}{2}\right)} + \left(\frac{1}{8} \cdot x + t\right) \]
   +-commutative [<=] 0.01	\[ z \cdot \left(-1 \cdot \frac{y}{2}\right) + \color{blue}{\left(t + \frac{1}{8} \cdot x\right)} \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(z, -1 \cdot \frac{y}{2}, t + \frac{1}{8} \cdot x\right)} \]
   associate-*r/ [=>] 0	\[ \mathsf{fma}\left(z, \color{blue}{\frac{-1 \cdot y}{2}}, t + \frac{1}{8} \cdot x\right) \]
   associate-/l* [=>] 0.06	\[ \mathsf{fma}\left(z, \color{blue}{\frac{-1}{\frac{2}{y}}}, t + \frac{1}{8} \cdot x\right) \]
   associate-/r/ [=>] 0	\[ \mathsf{fma}\left(z, \color{blue}{\frac{-1}{2} \cdot y}, t + \frac{1}{8} \cdot x\right) \]
   metadata-eval [=>] 0	\[ \mathsf{fma}\left(z, \color{blue}{-0.5} \cdot y, t + \frac{1}{8} \cdot x\right) \]
   +-commutative [=>] 0	\[ \mathsf{fma}\left(z, -0.5 \cdot y, \color{blue}{\frac{1}{8} \cdot x + t}\right) \]
   fma-def [=>] 0	\[ \mathsf{fma}\left(z, -0.5 \cdot y, \color{blue}{\mathsf{fma}\left(\frac{1}{8}, x, t\right)}\right) \]
   metadata-eval [=>] 0	\[ \mathsf{fma}\left(z, -0.5 \cdot y, \mathsf{fma}\left(\color{blue}{0.125}, x, t\right)\right) \]

3. Final simplification 0

   \[\leadsto \mathsf{fma}\left(z, -0.5 \cdot y, \mathsf{fma}\left(0.125, x, t\right)\right) \]

Alternatives

Alternative 1
Error	0.01%
Cost	7040

\[0.125 \cdot x - \mathsf{fma}\left(y \cdot 0.5, z, -t\right) \]

Alternative 2
Error	22.83%
Cost	1500

\[\begin{array}{l} t_1 := t + 0.125 \cdot x\\ t_2 := -0.5 \cdot \left(z \cdot y\right)\\ t_3 := t + t_2\\ \mathbf{if}\;z \leq -4 \cdot 10^{-42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-13}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+126}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+181}:\\ \;\;\;\;0.125 \cdot x + t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	23.59%
Cost	1243

\[\begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-44} \lor \neg \left(z \leq 1.12 \cdot 10^{-69}\right) \land \left(z \leq 1.02 \cdot 10^{-12} \lor \neg \left(z \leq 2 \cdot 10^{+112}\right) \land \left(z \leq 1.65 \cdot 10^{+126} \lor \neg \left(z \leq 2.05 \cdot 10^{+194}\right)\right)\right):\\ \;\;\;\;t + -0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]

Alternative 4
Error	47.14%
Cost	1116

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+83}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq -370000000:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-47}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-181}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;t \leq 45:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+50}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+150}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 5
Error	46.5%
Cost	984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+83}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq -30000000:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-60}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1950000:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+51}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+150}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 6
Error	18.73%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \cdot y \leq -5.5 \cdot 10^{+107} \lor \neg \left(z \cdot y \leq 7.2 \cdot 10^{+143}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]

Alternative 7
Error	0.01%
Cost	704

\[t + \left(0.125 \cdot x - z \cdot \frac{y}{2}\right) \]

Alternative 8
Error	62.62%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (+ (/ x 8.0) t) (* (/ z 2.0) y))

  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))