Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 3.13% → 3.13%

Time: 8.3s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x, double y, double z, double t) {
	return fma((y - x), (z / t), x);
}

Julia

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

↓

function code(x, y, z, t)
	return fma(Float64(y - x), Float64(z / t), x)
end

Wolfram

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

↓

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

Target

Original	3.13%
Target	3.36%
Herbie	3.13%

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} < -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

1. Initial program 3.13

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

2. Simplified 3.13

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]
   Proof

   [Start] 3.13	\[ x + \left(y - x\right) \cdot \frac{z}{t} \]
   +-commutative [=>] 3.13	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t} + x} \]
   fma-def [=>] 3.13	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

3. Final simplification 3.13

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

Alternatives

Alternative 1
Error	25.54%
Cost	1228

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+28}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	36.66%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{-x}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 3
Error	36.32%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;-\frac{x \cdot z}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 4
Error	36.35%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 5
Error	28.59%
Cost	977

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-191}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-217} \lor \neg \left(x \leq 10^{-61}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \end{array} \]

Alternative 6
Error	7.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{+18} \lor \neg \left(\frac{z}{t} \leq 10^{+28}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 7
Error	7.37%
Cost	968

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{+23}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]

Alternative 8
Error	36.83%
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-86} \lor \neg \left(\frac{z}{t} \leq 4 \cdot 10^{-105}\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	36.77%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 10
Error	3.13%
Cost	576

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

Alternative 11
Error	49.88%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

  (+ x (* (- y x) (/ z t))))