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

Average Error: 11.5 → 2.2

Time: 11.9s

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	11.5
Target	2.2
Herbie	2.2

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

Derivation

1. Initial program 11.5

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

2. Simplified 2.2

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

   [Start] 11.5	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
   associate-*r/ [<=] 2.2	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]
   sub-neg [=>] 2.2	\[ x \cdot \frac{\color{blue}{y + \left(-z\right)}}{t - z} \]
   +-commutative [=>] 2.2	\[ x \cdot \frac{\color{blue}{\left(-z\right) + y}}{t - z} \]
   neg-sub0 [=>] 2.2	\[ x \cdot \frac{\color{blue}{\left(0 - z\right)} + y}{t - z} \]
   associate-+l- [=>] 2.2	\[ x \cdot \frac{\color{blue}{0 - \left(z - y\right)}}{t - z} \]
   sub0-neg [=>] 2.2	\[ x \cdot \frac{\color{blue}{-\left(z - y\right)}}{t - z} \]
   neg-mul-1 [=>] 2.2	\[ x \cdot \frac{\color{blue}{-1 \cdot \left(z - y\right)}}{t - z} \]
   sub-neg [=>] 2.2	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{t + \left(-z\right)}} \]
   +-commutative [=>] 2.2	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(-z\right) + t}} \]
   neg-sub0 [=>] 2.2	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(0 - z\right)} + t} \]
   associate-+l- [=>] 2.2	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{0 - \left(z - t\right)}} \]
   sub0-neg [=>] 2.2	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-\left(z - t\right)}} \]
   neg-mul-1 [=>] 2.2	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-1 \cdot \left(z - t\right)}} \]
   times-frac [=>] 2.2	\[ x \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{z - y}{z - t}\right)} \]
   metadata-eval [=>] 2.2	\[ x \cdot \left(\color{blue}{1} \cdot \frac{z - y}{z - t}\right) \]
   *-lft-identity [=>] 2.2	\[ x \cdot \color{blue}{\frac{z - y}{z - t}} \]

3. Final simplification 2.2

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

Alternatives

Alternative 1
Error	18.6
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ t_2 := x \cdot \frac{y - z}{t}\\ t_3 := y \cdot \frac{x}{t - z}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-90}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+116}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	18.6
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ t_2 := y \cdot \frac{x}{t - z}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{+143}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 42000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 3
Error	18.5
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ t_2 := y \cdot \frac{x}{t - z}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 4
Error	18.6
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{\frac{z - t}{z}}\\ t_2 := y \cdot \frac{x}{t - z}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 5
Error	18.5
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{\frac{z - t}{z}}\\ t_2 := \frac{y}{\frac{t - z}{x}}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 6
Error	24.7
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	24.8
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{y}{\frac{z}{-x}}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	18.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-111} \lor \neg \left(z \leq 3.4 \cdot 10^{-87}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \end{array} \]

Alternative 9
Error	16.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-49} \lor \neg \left(z \leq 2 \cdot 10^{+37}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 10
Error	25.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	24.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	39.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(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)))