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

?

Percentage Accurate: 95.6% → 96.0%
Time: 17.2s
Precision: binary64
Cost: 7232

?

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
↓
\[x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{\frac{\frac{t}{z}}{y}}{3}\right) \]
(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))
↓
(FPCore (x y z t)
 :precision binary64
 (+ x (fma -0.3333333333333333 (/ y z) (/ (/ (/ t z) y) 3.0))))
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) {
	return x + fma(-0.3333333333333333, (y / z), (((t / z) / y) / 3.0));
}

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)
	return Float64(x + fma(-0.3333333333333333, Float64(y / z), Float64(Float64(Float64(t / z) / y) / 3.0)))
end

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_] := N[(x + N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + N[(N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{\frac{\frac{t}{z}}{y}}{3}\right)

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 16 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Target

Original95.6%
Target96.0%
Herbie96.0%
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation?

  1. Initial program 94.9%

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

  2. Simplified98.0%

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

    [Start]94.9%

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

    associate-+l- [=>]94.9%

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

    sub-neg [=>]94.9%

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

    sub-neg [=>]94.9%

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

    distribute-neg-in [=>]94.9%

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

    distribute-neg-frac [=>]94.9%

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

    neg-mul-1 [=>]94.9%

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

    *-commutative [=>]94.9%

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

    times-frac [=>]95.0%

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

    remove-double-neg [=>]95.0%

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

    fma-def [=>]95.0%

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

    metadata-eval [=>]95.0%

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

    associate-*l* [=>]95.0%

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

    associate-/r* [=>]98.0%

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

    associate-/l/ [<=]98.0%

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

  3. Final simplification98.0%

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

Alternatives

Alternative 1
Accuracy96.0%
Cost7232
\[x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{\frac{\frac{t}{z}}{y}}{3}\right) \]
Alternative 2
Accuracy80.5%
Cost1096
\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -4 \cdot 10^{+22}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-33}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y} - y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]
Alternative 3
Accuracy96.0%
Cost1088
\[\left(x + \frac{-1}{\frac{z \cdot 3}{y}}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]
Alternative 4
Accuracy96.0%
Cost960
\[\frac{\frac{t}{z \cdot 3}}{y} + \left(x - \frac{y}{z \cdot 3}\right) \]
Alternative 5
Accuracy88.3%
Cost840
\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+43}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333 \cdot y}{z}\\ \end{array} \]
Alternative 6
Accuracy91.9%
Cost840
\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+39}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;x + \frac{t \cdot \frac{0.3333333333333333}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333 \cdot y}{z}\\ \end{array} \]
Alternative 7
Accuracy92.0%
Cost840
\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+39}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{z}}{3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333 \cdot y}{z}\\ \end{array} \]
Alternative 8
Accuracy95.3%
Cost704
\[x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z} \]
Alternative 9
Accuracy95.3%
Cost704
\[x + -0.3333333333333333 \cdot \frac{y - \frac{t}{y}}{z} \]
Alternative 10
Accuracy48.3%
Cost585
\[\begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+80} \lor \neg \left(y \leq 1.45 \cdot 10^{+50}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 11
Accuracy48.3%
Cost585
\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+80} \lor \neg \left(y \leq 3.7 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{\frac{y}{-3}}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 12
Accuracy48.3%
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+80}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]
Alternative 13
Accuracy48.3%
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+80}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{z}\\ \end{array} \]
Alternative 14
Accuracy64.1%
Cost448
\[x + y \cdot \frac{-0.3333333333333333}{z} \]
Alternative 15
Accuracy64.1%
Cost448
\[x - \frac{y}{z} \cdot 0.3333333333333333 \]
Alternative 16
Accuracy30.2%
Cost64
\[x \]

Reproduce?

herbie shell --seed 2023263 
(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))))