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

Average Error: 1.4 → 0.5

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{if}\;y \leq -8.5196827904596 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.7040622508066614 \cdot 10^{-90}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a - t}\right) - \frac{y \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z t) (- a t)) x)))
   (if (<= y -8.5196827904596e+17)
     t_1
     (if (<= y 3.7040622508066614e-90)
       (- (+ x (/ (* y z) (- a t))) (/ (* y t) (- a t)))
       t_1))))

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 t_1 = fma(y, ((z - t) / (a - t)), x);
	double tmp;
	if (y <= -8.5196827904596e+17) {
		tmp = t_1;
	} else if (y <= 3.7040622508066614e-90) {
		tmp = (x + ((y * z) / (a - t))) - ((y * t) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(z - t) / Float64(a - t)), x)
	tmp = 0.0
	if (y <= -8.5196827904596e+17)
		tmp = t_1;
	elseif (y <= 3.7040622508066614e-90)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / Float64(a - t))) - Float64(Float64(y * t) / Float64(a - t)));
	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[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -8.5196827904596e+17], t$95$1, If[LessEqual[y, 3.7040622508066614e-90], N[(N[(x + N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.4
Target	0.4
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -851968279045959940 or 3.70406225080666137e-90 < y

   1. Initial program 0.6

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

   2. Simplified 0.6

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

   if -851968279045959940 < y < 3.70406225080666137e-90

   1. Initial program 2.4

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

   2. Simplified 2.4

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

   3. Taylor expanded in y around 0 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5196827904596 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;y \leq 3.7040622508066614 \cdot 10^{-90}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a - t}\right) - \frac{y \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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