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

Average Error: 10.9 → 0.7

Time: 12.3s

Precision: binary64

Cost: 7108

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1098577455461332 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;y \leq 2.807804593935509 \cdot 10^{-240}:\\ \;\;\;\;x + \frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array} \]

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 -2.1098577455461332e-10)
   (fma y (/ (- z t) (- z a)) x)
   (if (<= y 2.807804593935509e-240)
     (+ x (* (/ 1.0 (- z a)) (* y (- z t))))
     (+ x (/ y (/ (- z a) (- z t)))))))

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 <= -2.1098577455461332e-10) {
		tmp = fma(y, ((z - t) / (z - a)), x);
	} else if (y <= 2.807804593935509e-240) {
		tmp = x + ((1.0 / (z - a)) * (y * (z - t)));
	} else {
		tmp = x + (y / ((z - a) / (z - t)));
	}
	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 <= -2.1098577455461332e-10)
		tmp = fma(y, Float64(Float64(z - t) / Float64(z - a)), x);
	elseif (y <= 2.807804593935509e-240)
		tmp = Float64(x + Float64(Float64(1.0 / Float64(z - a)) * Float64(y * Float64(z - t))));
	else
		tmp = Float64(x + Float64(y / Float64(Float64(z - a) / Float64(z - t))));
	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, -2.1098577455461332e-10], N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.807804593935509e-240], N[(x + N[(N[(1.0 / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(z - a), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	10.9
Target	1.2
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if y < -2.10985774554613321e-10

   1. Initial program 21.7

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)} \]
      Proof
      (fma.f64 y (/.f64 (-.f64 z t) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) x)): 2 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))) x): 66 points increase in error, 18 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

   if -2.10985774554613321e-10 < y < 2.80780459393550879e-240

   1. Initial program 0.3

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

   2. Applied egg-rr 0.4

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

   if 2.80780459393550879e-240 < y

   1. Initial program 12.6

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

   2. Simplified 1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)} \]
      Proof
      (fma.f64 y (/.f64 (-.f64 z t) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) x)): 2 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))) x): 66 points increase in error, 18 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. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\frac{y}{\frac{z - a}{z - t}} + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1098577455461332 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;y \leq 2.807804593935509 \cdot 10^{-240}:\\ \;\;\;\;x + \frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.9
Cost	1436

\[\begin{array}{l} \mathbf{if}\;a \leq -9.286669380145274 \cdot 10^{+111}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-188}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-200}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z}\\ \mathbf{elif}\;a \leq 2.0561653431422132 \cdot 10^{-70}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 2.2561924771970795 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;a \leq 21727896154942992000:\\ \;\;\;\;y - \frac{t}{\frac{z}{y}}\\ \mathbf{elif}\;a \leq 3.726345723042559 \cdot 10^{+139}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{-\frac{a}{z}}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	1096

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{if}\;y \leq -2.1098577455461332 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.807804593935509 \cdot 10^{-240}:\\ \;\;\;\;x + \frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.8
Cost	972

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{if}\;a \leq -1.4595821664672595 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.6440420891395505 \cdot 10^{-146}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;a \leq 2.908115713570543 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	1.5
Cost	968

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.807804593935509 \cdot 10^{-240}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	10.9
Cost	840

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8643784894582903 \cdot 10^{-82}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	18.5
Cost	712

\[\begin{array}{l} t_1 := x - y \cdot \frac{z}{a}\\ \mathbf{if}\;a \leq -1.9857350767445347 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.726345723042559 \cdot 10^{+139}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-143}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 4.8643784894582903 \cdot 10^{-82}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 8
Error	1.2
Cost	704

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

Alternative 9
Error	21.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{-167}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-239}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Error	27.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+180}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.187195211287866 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 11
Error	28.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022291 
(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))))