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

Average Error: 1.4 → 1.3

Time: 6.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -7.319216852213493 \cdot 10^{+57}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;y \leq 5.547445071430195 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -7.319216852213493e+57)
   (+ x (* (- z t) (/ y (- z a))))
   (if (<= y 5.547445071430195e-141)
     (+ x (/ (* y (- z t)) (- z a)))
     (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) {
	double tmp;
	if (y <= -7.319216852213493e+57) {
		tmp = x + ((z - t) * (y / (z - a)));
	} else if (y <= 5.547445071430195e-141) {
		tmp = x + ((y * (z - t)) / (z - a));
	} else {
		tmp = fma(y, ((z - t) / (z - a)), x);
	}
	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 (y <= -7.319216852213493e+57)
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / Float64(z - a))));
	elseif (y <= 5.547445071430195e-141)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)));
	else
		tmp = fma(y, Float64(Float64(z - t) / Float64(z - a)), x);
	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[y, -7.319216852213493e+57], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.547445071430195e-141], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

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	1.2
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if y < -7.3192168522134934e57

   1. Initial program 0.4

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

   2. Taylor expanded in y around 0 29.1

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

   3. Applied *-un-lft-identity_binary64 29.1

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

   4. Applied times-frac_binary64 3.3

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

   5. Simplified 3.3

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

   if -7.3192168522134934e57 < y < 5.5474450714301953e-141

   1. Initial program 2.1

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

   2. Taylor expanded in y around 0 0.9

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

   if 5.5474450714301953e-141 < y

   1. Initial program 0.8

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

   2. Simplified 0.8

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.319216852213493 \cdot 10^{+57}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;y \leq 5.547445071430195 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \end{array} \]

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, A"
  :precision binary64

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

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