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

Average Error: 2.19% → 1.52%

Time: 13.2s

Precision: binary64

Cost: 9292

Math

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

↓

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+62}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (<= t_1 -1e+37)
     (+ x (/ (- z t) (/ (- a t) y)))
     (if (<= t_1 5e+62)
       (+ x (/ (* y (- z t)) (- a t)))
       (if (<= t_1 2e+301) t_1 (fma (- z t) (/ y (- a t)) x))))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (a - t)));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (t_1 <= -1e+37) {
		tmp = x + ((z - t) / ((a - t) / y));
	} else if (t_1 <= 5e+62) {
		tmp = x + ((y * (z - t)) / (a - t));
	} else if (t_1 <= 2e+301) {
		tmp = t_1;
	} else {
		tmp = fma((z - t), (y / (a - t)), x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (t_1 <= -1e+37)
		tmp = Float64(x + Float64(Float64(z - t) / Float64(Float64(a - t) / y)));
	elseif (t_1 <= 5e+62)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(a - t)));
	elseif (t_1 <= 2e+301)
		tmp = t_1;
	else
		tmp = fma(Float64(z - t), Float64(y / Float64(a - t)), x);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+37], N[(x + N[(N[(z - t), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+62], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], t$95$1, N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Target

Original	2.19%
Target	0.84%
Herbie	1.52%

\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))) < -9.99999999999999954e36

   1. Initial program 2.43

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

   2. Applied egg-rr 2.38

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

   if -9.99999999999999954e36 < (+.f64 x (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))) < 5.00000000000000029e62

   1. Initial program 1.38

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

   2. Simplified 1.67

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

      [Start] 1.38	\[ x + y \cdot \frac{z - t}{a - t} \]
      associate-*r/ [=>] 1.67	\[ x + \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

   if 5.00000000000000029e62 < (+.f64 x (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))) < 2.00000000000000011e301

   1. Initial program 0.12

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

   if 2.00000000000000011e301 < (+.f64 x (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))))

   1. Initial program 45.07

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

   2. Simplified 4.84

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

      [Start] 45.07	\[ x + y \cdot \frac{z - t}{a - t} \]
      +-commutative [=>] 45.07	\[ \color{blue}{y \cdot \frac{z - t}{a - t} + x} \]
      associate-*r/ [=>] 23.37	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} + x \]
      associate-*l/ [<=] 4.85	\[ \color{blue}{\frac{y}{a - t} \cdot \left(z - t\right)} + x \]
      *-commutative [=>] 4.85	\[ \color{blue}{\left(z - t\right) \cdot \frac{y}{a - t}} + x \]
      fma-def [=>] 4.84	\[ \color{blue}{\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 1.52

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \cdot \frac{z - t}{a - t} \leq -1 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;x + y \cdot \frac{z - t}{a - t} \leq 5 \cdot 10^{+62}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + y \cdot \frac{z - t}{a - t} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.83%
Cost	3148

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+62}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{z - t}{\frac{1}{y}}}{a - t}\\ \end{array} \]

Alternative 2
Error	33.85%
Cost	1504

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{-91}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-136}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{-268}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-193}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-159}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Error	33.44%
Cost	1116

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{-91}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -4.3 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-136}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-292}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-268}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4
Error	22.64%
Cost	1104

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-19}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-32}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+79}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+93}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Error	2.32%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-152} \lor \neg \left(t \leq 7 \cdot 10^{-7}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \end{array} \]

Alternative 6
Error	32.91%
Cost	852

\[\begin{array}{l} t_1 := y \cdot \frac{z}{a}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-137}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-290}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7
Error	33.06%
Cost	852

\[\begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-134}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-215}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-292}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-268}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8
Error	32.9%
Cost	852

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-137}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-290}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-268}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 9
Error	32.94%
Cost	852

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-136}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -1.12 \cdot 10^{-214}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-292}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-268}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10
Error	17.31%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-18} \lor \neg \left(t \leq 1.8 \cdot 10^{-30}\right):\\ \;\;\;\;x + \left(y - \frac{y}{\frac{t}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 11
Error	21.85%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-16}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+27}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12
Error	2.19%
Cost	704

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

Alternative 13
Error	30.91%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+144}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	42.36%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-224}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	45.11%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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