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

Average Error: 11.6 → 2.2

Time: 9.3s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

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 * ((y - z) / (t - z));
}

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 * ((y - z) / (t - z))
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 * ((y - z) / (t - z));
}

Python

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

↓

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

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(y - z) / Float64(t - z)))
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 * ((y - z) / (t - z));
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[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	11.6
Target	2.1
Herbie	2.2

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

Derivation

1. Initial program 11.6

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

2. Simplified 2.2

   \[\leadsto \color{blue}{x \cdot \frac{y - z}{t - z}} \]
   Proof
   (*.f64 x (/.f64 (-.f64 y z) (-.f64 t z))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))): 74 points increase in error, 29 points decrease in error

3. Final simplification 2.2

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

Alternatives

Alternative 1
Error	16.5
Cost	1108

\[\begin{array}{l} t_1 := \frac{x}{1 - \frac{t}{z}}\\ t_2 := x \cdot \frac{y}{t - z}\\ \mathbf{if}\;z \leq -2.45 \cdot 10^{+123}:\\ \;\;\;\;x - x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -5500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	20.8
Cost	976

\[\begin{array}{l} t_1 := x \cdot \frac{y}{t - z}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+64}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-26}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	16.8
Cost	976

\[\begin{array}{l} t_1 := x \cdot \frac{y}{t - z}\\ t_2 := x - x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-26}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	16.9
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{\frac{t}{y - z}}\\ t_2 := x - x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 150000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{t}{z}}\\ \end{array} \]

Alternative 5
Error	20.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+65}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	25.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -60:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	24.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	24.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	39.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))