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

Average Error: 0.02% → 0.01%

Time: 3.2s

Precision: binary64

Cost: 6976

Math

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

↓

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

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) (* 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), (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 Float64(fma(z, Float64(-0.5 * y), Float64(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[(N[(z * N[(-0.5 * y), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Target

Original	0.02%
Target	0.01%
Herbie	0.01%

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

Derivation

1. Initial program 0.02

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

2. Simplified 0.01

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

   [Start] 0.02	\[ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   associate-+l- [=>] 0.02	\[ \color{blue}{\frac{1}{8} \cdot x - \left(\frac{y \cdot z}{2} - t\right)} \]
   sub-neg [=>] 0.02	\[ \color{blue}{\frac{1}{8} \cdot x + \left(-\left(\frac{y \cdot z}{2} - t\right)\right)} \]
   neg-mul-1 [=>] 0.02	\[ \frac{1}{8} \cdot x + \color{blue}{-1 \cdot \left(\frac{y \cdot z}{2} - t\right)} \]
   *-commutative [=>] 0.02	\[ \frac{1}{8} \cdot x + \color{blue}{\left(\frac{y \cdot z}{2} - t\right) \cdot -1} \]
   cancel-sign-sub [<=] 0.02	\[ \color{blue}{\frac{1}{8} \cdot x - \left(-\left(\frac{y \cdot z}{2} - t\right)\right) \cdot -1} \]
   *-commutative [=>] 0.02	\[ \frac{1}{8} \cdot x - \color{blue}{-1 \cdot \left(-\left(\frac{y \cdot z}{2} - t\right)\right)} \]
   cancel-sign-sub-inv [=>] 0.02	\[ \color{blue}{\frac{1}{8} \cdot x + \left(--1\right) \cdot \left(-\left(\frac{y \cdot z}{2} - t\right)\right)} \]
   metadata-eval [=>] 0.02	\[ \frac{1}{8} \cdot x + \color{blue}{1} \cdot \left(-\left(\frac{y \cdot z}{2} - t\right)\right) \]
   neg-sub0 [=>] 0.02	\[ \frac{1}{8} \cdot x + 1 \cdot \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - t\right)\right)} \]
   associate-+l- [<=] 0.02	\[ \frac{1}{8} \cdot x + 1 \cdot \color{blue}{\left(\left(0 - \frac{y \cdot z}{2}\right) + t\right)} \]
   neg-sub0 [<=] 0.02	\[ \frac{1}{8} \cdot x + 1 \cdot \left(\color{blue}{\left(-\frac{y \cdot z}{2}\right)} + t\right) \]
   distribute-lft-in [=>] 0.02	\[ \frac{1}{8} \cdot x + \color{blue}{\left(1 \cdot \left(-\frac{y \cdot z}{2}\right) + 1 \cdot t\right)} \]
   *-lft-identity [=>] 0.02	\[ \frac{1}{8} \cdot x + \left(1 \cdot \left(-\frac{y \cdot z}{2}\right) + \color{blue}{t}\right) \]
   associate-+r+ [=>] 0.02	\[ \color{blue}{\left(\frac{1}{8} \cdot x + 1 \cdot \left(-\frac{y \cdot z}{2}\right)\right) + t} \]

3. Taylor expanded in x around 0 0.02

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

4. Simplified 0.01

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

   [Start] 0.02	\[ \left(-0.5 \cdot \left(y \cdot z\right) + 0.125 \cdot x\right) + t \]
   *-commutative [=>] 0.02	\[ \left(\color{blue}{\left(y \cdot z\right) \cdot -0.5} + 0.125 \cdot x\right) + t \]
   *-commutative [=>] 0.02	\[ \left(\color{blue}{\left(z \cdot y\right)} \cdot -0.5 + 0.125 \cdot x\right) + t \]
   associate-*r* [<=] 0.01	\[ \left(\color{blue}{z \cdot \left(y \cdot -0.5\right)} + 0.125 \cdot x\right) + t \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot -0.5, 0.125 \cdot x\right)} + t \]
   *-commutative [=>] 0.01	\[ \mathsf{fma}\left(z, \color{blue}{-0.5 \cdot y}, 0.125 \cdot x\right) + t \]

5. Final simplification 0.01

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

Alternatives

Alternative 1
Error	13.29%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-41} \lor \neg \left(x \leq 3000000000000\right):\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;t + -0.5 \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 2
Error	0.01%
Cost	704

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

Alternative 3
Error	27.63%
Cost	320

\[0.125 \cdot x + t \]

Reproduce

herbie shell --seed 2023121 
(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))