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

Average Error: 1.3 → 1.4

Time: 8.8s

Precision: binary64

Cost: 7240

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+70}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \end{array} \]

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 -1e+70)
   (+ x (* y (/ (- z t) (- z a))))
   (if (<= y 2e+40)
     (+ x (/ (* y (- z t)) (- z a)))
     (fma (- z t) (/ y (- 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 <= -1e+70) {
		tmp = x + (y * ((z - t) / (z - a)));
	} else if (y <= 2e+40) {
		tmp = x + ((y * (z - t)) / (z - a));
	} else {
		tmp = fma((z - t), (y / (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 <= -1e+70)
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))));
	elseif (y <= 2e+40)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)));
	else
		tmp = fma(Float64(z - t), Float64(y / 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, -1e+70], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+40], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Target

Original	1.3
Target	1.2
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000007e70

   1. Initial program 0.8

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

   if -1.00000000000000007e70 < y < 2.00000000000000006e40

   1. Initial program 1.7

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

   2. Simplified 1.1

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]
      Proof
      (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))))): 17 points increase in error, 43 points decrease in error

   if 2.00000000000000006e40 < y

   1. Initial program 0.6

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

   2. Simplified 3.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)} \]
      Proof
      (fma.f64 (-.f64 z t) (/.f64 y (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 z t) (/.f64 y (-.f64 z a))) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 z t) y) (-.f64 z a))) x): 39 points increase in error, 23 points decrease in error
      (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 z t) (-.f64 z a)) y)) x): 17 points increase in error, 43 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 y (/.f64 (-.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

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+70}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	4124

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{z - a}\\ t_2 := \frac{z - t}{z - a}\\ t_3 := x + t \cdot \frac{y}{a}\\ t_4 := \frac{-y}{\frac{z - a}{t}}\\ \mathbf{if}\;t_2 \leq -20000000000:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -4 \cdot 10^{-145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0.9999999999:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(-t\right)\\ \end{array} \]

Alternative 2
Error	9.4
Cost	3608

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{-y}{\frac{z - a}{t}}\\ \mathbf{if}\;t_1 \leq -20000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -8 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{y \cdot z}{z - a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(-t\right)\\ \end{array} \]

Alternative 3
Error	12.7
Cost	3092

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-145}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{-y}{\frac{z - a}{t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(-t\right)\\ \end{array} \]

Alternative 4
Error	0.9
Cost	1348

\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{z - a}\\ \end{array} \]

Alternative 5
Error	0.9
Cost	968

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-90}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	14.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+19}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+15}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Error	14.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+25}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+17}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 8
Error	20.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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