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

Average Error: 12.0 → 2.0

Time: 12.6s

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	12.0
Target	2.0
Herbie	2.0

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

Derivation

1. Initial program 12.0

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

2. Simplified 2.0

   \[\leadsto \color{blue}{x \cdot \frac{y - z}{t - z}} \]
   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))): 63 points increase in error, 30 points decrease in error

3. Final simplification 2.0

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

Alternatives

Alternative 1
Error	17.4
Cost	1372

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{z - y}}\\ t_2 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t \leq -1.32 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{+18}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -210000000:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \end{array} \]

Alternative 2
Error	17.4
Cost	1372

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{z - y}}\\ \mathbf{if}\;t \leq -7 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{elif}\;t \leq -14500000000:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \end{array} \]

Alternative 3
Error	17.5
Cost	1372

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;t \leq -1.28 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{elif}\;t \leq -440000000:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \end{array} \]

Alternative 4
Error	17.5
Cost	1372

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;t \leq -2.65 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{\frac{t - z}{-z}}\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{elif}\;t \leq -170000000:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \end{array} \]

Alternative 5
Error	24.3
Cost	1240

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{x}{t}\\ t_2 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{-\frac{z}{y}}\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{+195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+143}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \end{array} \]

Alternative 6
Error	24.1
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{-\frac{z}{y}}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{+193}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \end{array} \]

Alternative 7
Error	24.0
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{-\frac{z}{y}}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+196}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq -8.7 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \end{array} \]

Alternative 8
Error	24.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 340:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	17.3
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{z - y}}\\ t_2 := \frac{x}{\frac{t}{y - z}}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{+120}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t \leq -1.32 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -430000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	26.5
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 235:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	20.6
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+183}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-80}:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{elif}\;z \leq 52:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	26.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 44:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	25.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 54:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	39.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(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)))