Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Average Error: 16.56% → 2%

Time: 10.1s

Precision: binary64

Cost: 6976

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	16.56%
Target	1.81%
Herbie	2%

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

Derivation

1. Initial program 16.56

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

2. Simplified 2

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

   [Start] 16.56	\[ x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   +-commutative [=>] 16.56	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]
   associate-*r/ [<=] 2	\[ \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]
   fma-def [=>] 2	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)} \]

3. Final simplification 2

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

Alternatives

Alternative 1
Error	25.54%
Cost	1237

\[\begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+55}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+16} \lor \neg \left(z \leq 1.35 \cdot 10^{+48}\right) \land z \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 2
Error	24.88%
Cost	1172

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+52}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-21}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{y \cdot t}{z}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+53}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 3
Error	20.18%
Cost	1105

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+50} \lor \neg \left(z \leq 8 \cdot 10^{+121}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternative 4
Error	4.86%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-231} \lor \neg \left(x \leq 2.7 \cdot 10^{-264}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternative 5
Error	19.74%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-12} \lor \neg \left(z \leq 4.2 \cdot 10^{-25}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 6
Error	22.87%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.46 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-24}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Error	22.69%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+50}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 8
Error	1.81%
Cost	704

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

Alternative 9
Error	30.38%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+156}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+192}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	44.63%
Cost	64

\[x \]

Reproduce

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

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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