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

Average Error: 11.0 → 1.4

Time: 49.0s

Precision: binary64

Cost: 7240

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-286}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \end{array} \]

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 -2e-286)
   (+ x (/ y (/ (- z a) (- z t))))
   (if (<= y 7.2e+40)
     (+ x (/ (* y (- z t)) (- z a)))
     (fma (/ y (- z a)) (- 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 <= -2e-286) {
		tmp = x + (y / ((z - a) / (z - t)));
	} else if (y <= 7.2e+40) {
		tmp = x + ((y * (z - t)) / (z - a));
	} else {
		tmp = fma((y / (z - a)), (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 <= -2e-286)
		tmp = Float64(x + Float64(y / Float64(Float64(z - a) / Float64(z - t))));
	elseif (y <= 7.2e+40)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)));
	else
		tmp = fma(Float64(y / Float64(z - a)), 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, -2e-286], N[(x + N[(y / N[(N[(z - a), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+40], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]]]

Target

Original	11.0
Target	1.2
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if y < -2.0000000000000001e-286

   1. Initial program 11.2

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

   2. Simplified 1.3

      \[\leadsto \color{blue}{x + \frac{y}{\frac{z - a}{z - t}}} \]
      Proof
      (+.f64 x (/.f64 y (/.f64 (-.f64 z a) (-.f64 z t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)))): 44 points increase in error, 21 points decrease in error

   if -2.0000000000000001e-286 < y < 7.19999999999999993e40

   1. Initial program 0.6

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

   if 7.19999999999999993e40 < y

   1. Initial program 26.9

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

   2. Simplified 3.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
      Proof
      (fma.f64 (/.f64 y (-.f64 z a)) (-.f64 z t) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 y (-.f64 z a)) (-.f64 z t)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))) x): 39 points increase in error, 23 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-286}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1992

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z - t}}\\ t_2 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+278}:\\ \;\;\;\;x + t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.6
Cost	972

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+124}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+16}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 3
Error	15.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+124}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+16}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 4
Error	14.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+23}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 5
Error	14.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+24}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+15}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 6
Error	1.2
Cost	704

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

Alternative 7
Error	20.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+135}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	27.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+90}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+162}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 9
Error	29.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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))))