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

Average Error: 16.52% → 1.06%

Time: 12.5s

Precision: binary64

Cost: 7241

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-141} \lor \neg \left(y \leq 4 \cdot 10^{-100}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -7e-141) (not (<= y 4e-100)))
   (fma y (/ (- z t) (- a t)) x)
   (+ x (/ (* y (- z t)) (- a t)))))

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 tmp;
	if ((y <= -7e-141) || !(y <= 4e-100)) {
		tmp = fma(y, ((z - t) / (a - t)), x);
	} else {
		tmp = x + ((y * (z - t)) / (a - t));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -7e-141) || !(y <= 4e-100))
		tmp = fma(y, Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(a - t)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -7e-141], N[Not[LessEqual[y, 4e-100]], $MachinePrecision]], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	16.52%
Target	2.04%
Herbie	1.06%

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

Derivation

1. Split input into 2 regimes

2. if y < -7.0000000000000006e-141 or 4.0000000000000001e-100 < y

   1. Initial program 24.75

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

   2. Simplified 1.28

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

      [Start] 24.75	\[ x + \frac{y \cdot \left(z - t\right)}{a - t} \]
      +-commutative [=>] 24.75	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t} + x} \]
      associate-*r/ [<=] 1.29	\[ \color{blue}{y \cdot \frac{z - t}{a - t}} + x \]
      fma-def [=>] 1.28	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)} \]

   if -7.0000000000000006e-141 < y < 4.0000000000000001e-100

   1. Initial program 0.63

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.06

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-141} \lor \neg \left(y \leq 4 \cdot 10^{-100}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.71%
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{a - t}{z}}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+215}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-194}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 2
Error	13.09%
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{a - t}{z}}\\ t_2 := x - \frac{y}{\frac{a}{t} + -1}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-195}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 1.78 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	27.58%
Cost	976

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+88}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 4
Error	4.41%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-75} \lor \neg \left(z \leq 5 \cdot 10^{-242}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t} + -1}\\ \end{array} \]

Alternative 5
Error	31.31%
Cost	844

\[\begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-120}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-306}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-287}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 6
Error	21.62%
Cost	844

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+44}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-237}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Error	18.08%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+215} \lor \neg \left(t \leq 5.1 \cdot 10^{+176}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z}}\\ \end{array} \]

Alternative 8
Error	25.93%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-5} \lor \neg \left(a \leq 1.1 \cdot 10^{+89}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Error	25.73%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+88}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 10
Error	2.04%
Cost	704

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

Alternative 11
Error	31.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+49}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{+189}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	42.55%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-255}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-218}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	45.01%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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