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

Average Error: 11.2 → 0.6

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -166269836987335460:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;y \leq 4.180122896346843 \cdot 10^{-151}:\\ \;\;\;\;\frac{y \cdot z}{a - t} + \left(x - \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{1}{\frac{a - t}{z - t}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -166269836987335460.0)
   (fma y (/ (- z t) (- a t)) x)
   (if (<= y 4.180122896346843e-151)
     (+ (/ (* y z) (- a t)) (- x (/ (* y t) (- a t))))
     (+ x (* y (/ 1.0 (/ (- a t) (- z t))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -166269836987335460.0) {
		tmp = fma(y, ((z - t) / (a - t)), x);
	} else if (y <= 4.180122896346843e-151) {
		tmp = ((y * z) / (a - t)) + (x - ((y * t) / (a - t)));
	} else {
		tmp = x + (y * (1.0 / ((a - t) / (z - t))));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -166269836987335460.0)
		tmp = fma(y, Float64(Float64(z - t) / Float64(a - t)), x);
	elseif (y <= 4.180122896346843e-151)
		tmp = Float64(Float64(Float64(y * z) / Float64(a - t)) + Float64(x - Float64(Float64(y * t) / Float64(a - t))));
	else
		tmp = Float64(x + Float64(y * Float64(1.0 / Float64(Float64(a - t) / Float64(z - t)))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -166269836987335460.0], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 4.180122896346843e-151], N[(N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(y * t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(1.0 / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	11.2
Target	1.1
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if y < -166269836987335456

   1. Initial program 25.0

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

   2. Simplified 0.4

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

   if -166269836987335456 < y < 4.18012289634684304e-151

   1. Initial program 0.4

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

   2. Simplified 2.0

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

   3. Taylor expanded in z around 0 0.4

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

   if 4.18012289634684304e-151 < y

   1. Initial program 15.5

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

   2. Applied egg-rr 1.0

      \[\leadsto x + \color{blue}{y \cdot \frac{1}{\frac{a - t}{z - t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -166269836987335460:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;y \leq 4.180122896346843 \cdot 10^{-151}:\\ \;\;\;\;\frac{y \cdot z}{a - t} + \left(x - \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{1}{\frac{a - t}{z - t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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