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

Average Error: 1.3 → 1.2

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{z - a}\\ \mathbf{if}\;y \leq -5.0433371292086095 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{elif}\;y \leq 4.435070156250409 \cdot 10^{-230}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{t_1}^{2}}, \frac{z - t}{t_1}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (cbrt (- z a))))
   (if (<= y -5.0433371292086095e-118)
     (+ (/ y (/ (- z a) (- z t))) x)
     (if (<= y 4.435070156250409e-230)
       (+ x (/ (* y (- t z)) (- a z)))
       (fma (/ y (pow t_1 2.0)) (/ (- z t) t_1) 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 t_1 = cbrt((z - a));
	double tmp;
	if (y <= -5.0433371292086095e-118) {
		tmp = (y / ((z - a) / (z - t))) + x;
	} else if (y <= 4.435070156250409e-230) {
		tmp = x + ((y * (t - z)) / (a - z));
	} else {
		tmp = fma((y / pow(t_1, 2.0)), ((z - t) / t_1), 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)
	t_1 = cbrt(Float64(z - a))
	tmp = 0.0
	if (y <= -5.0433371292086095e-118)
		tmp = Float64(Float64(y / Float64(Float64(z - a) / Float64(z - t))) + x);
	elseif (y <= 4.435070156250409e-230)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / Float64(a - z)));
	else
		tmp = fma(Float64(y / (t_1 ^ 2.0)), Float64(Float64(z - t) / t_1), 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_] := Block[{t$95$1 = N[Power[N[(z - a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[y, -5.0433371292086095e-118], N[(N[(y / N[(N[(z - a), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 4.435070156250409e-230], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.3
Target	1.2
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if y < -5.04333712920860952e-118

   1. Initial program 0.7

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

   2. Applied egg-rr 0.6

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

   3. Applied egg-rr 0.7

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

   4. Applied egg-rr 0.6

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

   if -5.04333712920860952e-118 < y < 4.4350701562504092e-230

   1. Initial program 2.7

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

   2. Applied egg-rr 0.2

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

   if 4.4350701562504092e-230 < y

   1. Initial program 1.1

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

   2. Applied egg-rr 2.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{{\left(\sqrt[3]{z - a}\right)}^{2}}, \frac{z - t}{\sqrt[3]{z - a}}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.0433371292086095 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{elif}\;y \leq 4.435070156250409 \cdot 10^{-230}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{\left(\sqrt[3]{z - a}\right)}^{2}}, \frac{z - t}{\sqrt[3]{z - a}}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))))