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

Average Error: 1.4 → 0.7

Time: 13.5s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{if}\;y \leq -2.1098577455461332 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.807804593935509 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(z - t\right), \frac{1}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ y (/ (- z a) (- z t))))))
   (if (<= y -2.1098577455461332e-10)
     t_1
     (if (<= y 2.807804593935509e-240)
       (fma (* y (- z t)) (/ 1.0 (- z a)) x)
       t_1))))

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 t_1 = x + (y / ((z - a) / (z - t)));
	double tmp;
	if (y <= -2.1098577455461332e-10) {
		tmp = t_1;
	} else if (y <= 2.807804593935509e-240) {
		tmp = fma((y * (z - t)), (1.0 / (z - a)), x);
	} else {
		tmp = t_1;
	}
	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)
	t_1 = Float64(x + Float64(y / Float64(Float64(z - a) / Float64(z - t))))
	tmp = 0.0
	if (y <= -2.1098577455461332e-10)
		tmp = t_1;
	elseif (y <= 2.807804593935509e-240)
		tmp = fma(Float64(y * Float64(z - t)), Float64(1.0 / Float64(z - a)), x);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x + N[(y / N[(N[(z - a), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1098577455461332e-10], t$95$1, If[LessEqual[y, 2.807804593935509e-240], N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Target

Original	1.4
Target	1.2
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if y < -2.10985774554613321e-10 or 2.80780459393550879e-240 < y

   1. Initial program 1.0

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

   2. Applied egg-rr 0.9

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

   if -2.10985774554613321e-10 < y < 2.80780459393550879e-240

   1. Initial program 2.2

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

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(z - t\right), \frac{1}{z - a}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1098577455461332 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;y \leq 2.807804593935509 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(z - t\right), \frac{1}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{+134}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a - z}, x\right)\\ \end{array} \]

Alternative 2
Error	1.1
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a - z}{y}}\\ \end{array} \]

Alternative 3
Error	19.7
Cost	976

\[\begin{array}{l} t_1 := x - \frac{y}{\frac{a}{z}}\\ \mathbf{if}\;a \leq -1.9857350767445347 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.4393556939422858 \cdot 10^{-64}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 5.3816139490632374 \cdot 10^{-14}:\\ \;\;\;\;\frac{t}{\frac{a - z}{y}}\\ \mathbf{elif}\;a \leq 3.726345723042559 \cdot 10^{+139}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.3
Cost	972

\[\begin{array}{l} t_1 := \left(t - z\right) \cdot \frac{y}{a - z}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+203}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{elif}\;t \leq 10^{+250}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \frac{t}{z}\right)\\ \end{array} \]

Alternative 5
Error	12.1
Cost	840

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8643784894582903 \cdot 10^{-82}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	10.3
Cost	840

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8643784894582903 \cdot 10^{-82}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.5
Cost	712

\[\begin{array}{l} t_1 := x - \frac{y}{\frac{a}{z}}\\ \mathbf{if}\;a \leq -1.9857350767445347 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.726345723042559 \cdot 10^{+139}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-143}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.8482931920155947 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Error	21.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{-167}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-239}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Error	27.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+180}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.187195211287866 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 11
Error	28.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022291 
(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)))))