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

Average Error: 0.0 → 0.0

Time: 4.3s

Precision: binary64

Cost: 6784

Math

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

↓

\[\mathsf{fma}\left(z + y, x, -z\right) \]

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma (+ z y) x (- 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 fma((z + y), x, -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 fma(Float64(z + y), x, Float64(-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[(z + y), $MachinePrecision] * x + (-z)), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{\left(y + z\right) \cdot x + -1 \cdot z} \]

3. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, x, -z\right)} \]
   Proof
   (fma.f64 (+.f64 z y) x (neg.f64 z)): 0 points increase in error, 0 points decrease in error
   (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 y z)) x (neg.f64 z)): 0 points increase in error, 0 points decrease in error
   (fma.f64 (+.f64 y z) x (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 y z) x) (*.f64 -1 z))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(z + y, x, -z\right) \]

Alternatives

Alternative 1
Error	24.4
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+33}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-86}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-56}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+31}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 2
Error	13.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-86} \lor \neg \left(x \leq 2.4 \cdot 10^{-56}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Error	13.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-86} \lor \neg \left(x \leq 1.45 \cdot 10^{-55}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x - z\\ \end{array} \]

Alternative 5
Error	24.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-86}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-56}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

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

Alternative 7
Error	35.1
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))