Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3

Average Error: 0.0 → 0.0

Time: 2.2s

Precision: binary64

Cost: 448

Math

\[x \cdot y + \left(x - 1\right) \cdot z \]

↓

\[x \cdot \left(y + z\right) - z \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- x 1.0) z)))

↓

(FPCore (x y z) :precision binary64 (- (* x (+ y z)) z))

C

double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * z);
}

↓

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + ((x - 1.0d0) * z)
end function

↓

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

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * z);
}

↓

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

Python

def code(x, y, z):
	return (x * y) + ((x - 1.0) * z)

↓

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

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(x - 1.0) * z))
end

↓

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

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((x - 1.0) * z);
end

↓

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

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

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

Derivation

1. Initial program 0.0

   \[x \cdot y + \left(x - 1\right) \cdot z \]

2. Simplified 0.0

   \[\leadsto \color{blue}{x \cdot \left(y + z\right) - z} \]
   Proof
   (-.f64 (*.f64 x (+.f64 y z)) z): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x y) (*.f64 x z))) z): 1 points increase in error, 1 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (neg.f64 z))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) z)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 x z) (*.f64 (neg.f64 1) z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 x y) (Rewrite<= distribute-rgt-in_binary64 (*.f64 z (+.f64 x (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 x y) (*.f64 z (Rewrite<= sub-neg_binary64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 x y) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 x 1) z))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

   \[\leadsto x \cdot \left(y + z\right) - z \]

Alternatives

Alternative 1
Error	0.9
Cost	584

\[\begin{array}{l} t_0 := x \cdot \left(y + z\right)\\ \mathbf{if}\;x \leq -50000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;x \cdot y - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	35.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-205}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{-121}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Error	27.7
Cost	320

\[x \cdot \left(y + z\right) \]

Alternative 4
Error	39.4
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))