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

Average Error: 1.3 → 0.7

Time: 21.7s

Precision: binary64

Cost: 7240

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{a - z}, y, x\right)\\ \mathbf{if}\;y \leq -5400000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-164}:\\ \;\;\;\;x + \frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t z) (- a z)) y x)))
   (if (<= y -5400000000000.0)
     t_1
     (if (<= y 2e-164) (+ x (* (/ 1.0 (- z a)) (* y (- z t)))) t_1))))

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 t_1 = fma(((t - z) / (a - z)), y, x);
	double tmp;
	if (y <= -5400000000000.0) {
		tmp = t_1;
	} else if (y <= 2e-164) {
		tmp = x + ((1.0 / (z - a)) * (y * (z - t)));
	} else {
		tmp = t_1;
	}
	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)
	t_1 = fma(Float64(Float64(t - z) / Float64(a - z)), y, x)
	tmp = 0.0
	if (y <= -5400000000000.0)
		tmp = t_1;
	elseif (y <= 2e-164)
		tmp = Float64(x + Float64(Float64(1.0 / Float64(z - a)) * Float64(y * Float64(z - t))));
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[y, -5400000000000.0], t$95$1, If[LessEqual[y, 2e-164], N[(x + N[(N[(1.0 / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	1.3
Target	1.2
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if y < -5.4e12 or 1.99999999999999992e-164 < y

   1. Initial program 0.8

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

   2. Simplified 0.8

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

   if -5.4e12 < y < 1.99999999999999992e-164

   1. Initial program 2.0

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

   2. Applied egg-rr 0.5

      \[\leadsto x + \color{blue}{\frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)} \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	0.8
Cost	1348

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

Alternative 2
Error	0.7
Cost	1096

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-164}:\\ \;\;\;\;x + \frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	17.0
Cost	976

\[\begin{array}{l} t_1 := x + y \cdot \frac{t}{a}\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{-272}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{-279}:\\ \;\;\;\;x + \frac{t}{z} \cdot \left(-y\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-57}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	11.5
Cost	840

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{a} \cdot t + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	10.6
Cost	840

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z}\\ \mathbf{if}\;z \leq -1450000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-41}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	14.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1450000000000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-86}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Error	14.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3300000000000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{a} \cdot t + x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 8
Error	1.3
Cost	704

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

Alternative 9
Error	20.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Error	28.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(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)))))