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

Average Accuracy: 97.9% → 96.3%

Time: 12.8s

Precision: binary64

Cost: 7108

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{-194}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= a 2.5e-194)
   (fma (- z t) (/ y (- z a)) x)
   (+ x (* y (/ (- z t) (- z a))))))

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) {
	double tmp;
	if (a <= 2.5e-194) {
		tmp = fma((z - t), (y / (z - a)), x);
	} else {
		tmp = x + (y * ((z - t) / (z - a)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 2.5e-194)
		tmp = fma(Float64(z - t), Float64(y / Float64(z - a)), x);
	else
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[a, 2.5e-194], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	97.9%
Target	98.0%
Herbie	96.3%

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

Derivation

1. Split input into 2 regimes

2. if a < 2.5000000000000001e-194

   1. Initial program 97.3%

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

   2. Simplified 94.6%

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

      [Start] 97.3	\[ x + y \cdot \frac{z - t}{z - a} \]
      +-commutative [=>] 97.3	\[ \color{blue}{y \cdot \frac{z - t}{z - a} + x} \]
      associate-*r/ [=>] 82.8	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x \]
      associate-*l/ [<=] 94.6	\[ \color{blue}{\frac{y}{z - a} \cdot \left(z - t\right)} + x \]
      *-commutative [=>] 94.6	\[ \color{blue}{\left(z - t\right) \cdot \frac{y}{z - a}} + x \]
      fma-def [=>] 94.6	\[ \color{blue}{\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)} \]

   if 2.5000000000000001e-194 < a

   1. Initial program 98.7%

      \[x + y \cdot \frac{z - t}{z - a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{-194}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	80.7%
Cost	3156

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := x + \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;t_1 \leq -500000:\\ \;\;\;\;t \cdot \frac{-y}{z - a}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x + z \cdot \frac{y}{z - a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;y \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	80.7%
Cost	3156

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := x + \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;t_1 \leq -500000:\\ \;\;\;\;\frac{z - t}{\frac{z - a}{y}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x + z \cdot \frac{y}{z - a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;y \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	81.3%
Cost	2640

\[\begin{array}{l} t_1 := x + \frac{t}{\frac{a}{y}}\\ t_2 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+26}:\\ \;\;\;\;t \cdot \frac{-y}{z - a}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 1:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;y \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	97.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-81} \lor \neg \left(z \leq 5 \cdot 10^{-82}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{z - a}\\ \end{array} \]

Alternative 5
Accuracy	76.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+159}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -0.000102:\\ \;\;\;\;x - t \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+16}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 6
Accuracy	96.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{-194}:\\ \;\;\;\;x + \frac{z - t}{\frac{z - a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternative 7
Accuracy	77.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -490000000 \lor \neg \left(z \leq 7.5 \cdot 10^{+16}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 8
Accuracy	97.9%
Cost	704

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

Alternative 9
Accuracy	68.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -7800:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Accuracy	57.3%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-223}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	54.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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