Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Average Error: 12.1 → 2.4

Time: 8.6s

Precision: binary64

Cost: 576

Math

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

↓

\[x \cdot \frac{z - y}{z - t} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x (- y z)) (- t z)))

↓

(FPCore (x y z t) :precision binary64 (* x (/ (- z y) (- z t))))

C

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

↓

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * (y - z)) / (t - z)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * ((z - y) / (z - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * (y - z)) / (t - z);
}

↓

public static double code(double x, double y, double z, double t) {
	return x * ((z - y) / (z - t));
}

Python

def code(x, y, z, t):
	return (x * (y - z)) / (t - z)

↓

def code(x, y, z, t):
	return x * ((z - y) / (z - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(y - z)) / Float64(t - z))
end

↓

function code(x, y, z, t)
	return Float64(x * Float64(Float64(z - y) / Float64(z - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * (y - z)) / (t - z);
end

↓

function tmp = code(x, y, z, t)
	tmp = x * ((z - y) / (z - t));
end

Wolfram

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

↓

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

Target

Original	12.1
Target	2.5
Herbie	2.4

\[\frac{x}{\frac{t - z}{y - z}} \]

Derivation

1. Initial program 12.1

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

2. Simplified 2.4

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

   [Start] 12.1	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
   associate-*r/ [<=] 2.4	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]
   sub-neg [=>] 2.4	\[ x \cdot \frac{\color{blue}{y + \left(-z\right)}}{t - z} \]
   +-commutative [=>] 2.4	\[ x \cdot \frac{\color{blue}{\left(-z\right) + y}}{t - z} \]
   neg-sub0 [=>] 2.4	\[ x \cdot \frac{\color{blue}{\left(0 - z\right)} + y}{t - z} \]
   associate-+l- [=>] 2.4	\[ x \cdot \frac{\color{blue}{0 - \left(z - y\right)}}{t - z} \]
   sub0-neg [=>] 2.4	\[ x \cdot \frac{\color{blue}{-\left(z - y\right)}}{t - z} \]
   neg-mul-1 [=>] 2.4	\[ x \cdot \frac{\color{blue}{-1 \cdot \left(z - y\right)}}{t - z} \]
   sub-neg [=>] 2.4	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{t + \left(-z\right)}} \]
   +-commutative [=>] 2.4	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(-z\right) + t}} \]
   neg-sub0 [=>] 2.4	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(0 - z\right)} + t} \]
   associate-+l- [=>] 2.4	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{0 - \left(z - t\right)}} \]
   sub0-neg [=>] 2.4	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-\left(z - t\right)}} \]
   neg-mul-1 [=>] 2.4	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-1 \cdot \left(z - t\right)}} \]
   times-frac [=>] 2.4	\[ x \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{z - y}{z - t}\right)} \]
   metadata-eval [=>] 2.4	\[ x \cdot \left(\color{blue}{1} \cdot \frac{z - y}{z - t}\right) \]
   *-lft-identity [=>] 2.4	\[ x \cdot \color{blue}{\frac{z - y}{z - t}} \]

3. Final simplification 2.4

   \[\leadsto x \cdot \frac{z - y}{z - t} \]

Alternatives

Alternative 1
Error	22.7
Cost	1504

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{+177}:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{elif}\;t \leq -2.05 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-158}:\\ \;\;\;\;x - \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;t \leq 1.58 \cdot 10^{-124}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	23.2
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{z}\right)\\ t_2 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;t \leq -3.65 \cdot 10^{+220}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+168}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	23.2
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+216}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-154}:\\ \;\;\;\;x - \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+170}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	19.4
Cost	977

\[\begin{array}{l} t_1 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;y \leq -1.06 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-75}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+141} \lor \neg \left(y \leq 4.15 \cdot 10^{+210}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 5
Error	18.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-69} \lor \neg \left(z \leq 4 \cdot 10^{-88}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \end{array} \]

Alternative 6
Error	16.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-69} \lor \neg \left(z \leq 40000\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 7
Error	26.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 25000:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	25.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 40000:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	39.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (/ x (/ (- t z) (- y z)))

  (/ (* x (- y z)) (- t z)))