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

Average Error: 10.7 → 0.7

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.74052409511644 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;y \leq 2.8892448090824406 \cdot 10^{-235}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{z - a}\right) - \frac{y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{1}{z - a}}{\frac{1}{z - t}}, x\right)\\ \end{array} \]

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 -1.74052409511644e-97)
   (fma y (/ (- z t) (- z a)) x)
   (if (<= y 2.8892448090824406e-235)
     (- (+ x (/ (* y z) (- z a))) (/ (* y t) (- z a)))
     (fma y (/ (/ 1.0 (- z a)) (/ 1.0 (- z t))) 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 <= -1.74052409511644e-97) {
		tmp = fma(y, ((z - t) / (z - a)), x);
	} else if (y <= 2.8892448090824406e-235) {
		tmp = (x + ((y * z) / (z - a))) - ((y * t) / (z - a));
	} else {
		tmp = fma(y, ((1.0 / (z - a)) / (1.0 / (z - t))), x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.74052409511644e-97)
		tmp = fma(y, Float64(Float64(z - t) / Float64(z - a)), x);
	elseif (y <= 2.8892448090824406e-235)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / Float64(z - a))) - Float64(Float64(y * t) / Float64(z - a)));
	else
		tmp = fma(y, Float64(Float64(1.0 / Float64(z - a)) / Float64(1.0 / Float64(z - t))), x);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.74052409511644e-97], N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.8892448090824406e-235], N[(N[(x + N[(N[(y * z), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 / N[(z - a), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(z - t), $MachinePrecision]), $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	10.7
Target	1.2
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if y < -1.74052409511643997e-97

   1. Initial program 17.8

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

   2. Simplified 0.6

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

   if -1.74052409511643997e-97 < y < 2.88924480908244063e-235

   1. Initial program 0.2

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

   2. Simplified 2.6

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

   3. Applied egg-rr 2.7

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\sqrt[3]{\frac{z - t}{z - a}} \cdot {\left(\sqrt[3]{\frac{z - t}{z - a}}\right)}^{2}}, x\right) \]

   4. Taylor expanded in y around 0 0.2

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

   5. Simplified 0.2

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

   if 2.88924480908244063e-235 < y

   1. Initial program 11.5

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

   2. Simplified 1.1

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

   3. Applied egg-rr 1.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{{\left(\frac{z - a}{z - t}\right)}^{-1}}, x\right) \]

   4. Applied egg-rr 1.1

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.74052409511644 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;y \leq 2.8892448090824406 \cdot 10^{-235}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{z - a}\right) - \frac{y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{1}{z - a}}{\frac{1}{z - t}}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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