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

Average Error: 1.4 → 1.1

Time: 4.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9258428577443835 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{t - a}, x\right)\\ \mathbf{elif}\;y \leq 1.0514448438020326 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(z - t\right), \frac{1}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -3.9258428577443835e+52)
   (fma (- t z) (/ y (- t a)) x)
   (if (<= y 1.0514448438020326e-28)
     (fma (* y (- z t)) (/ 1.0 (- a t)) x)
     (+ x (/ y (/ (- a t) (- z 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 <= -3.9258428577443835e+52) {
		tmp = fma((t - z), (y / (t - a)), x);
	} else if (y <= 1.0514448438020326e-28) {
		tmp = fma((y * (z - t)), (1.0 / (a - t)), x);
	} else {
		tmp = x + (y / ((a - t) / (z - t)));
	}
	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)
	tmp = 0.0
	if (y <= -3.9258428577443835e+52)
		tmp = fma(Float64(t - z), Float64(y / Float64(t - a)), x);
	elseif (y <= 1.0514448438020326e-28)
		tmp = fma(Float64(y * Float64(z - t)), Float64(1.0 / Float64(a - t)), x);
	else
		tmp = Float64(x + Float64(y / Float64(Float64(a - t) / Float64(z - t))));
	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_] := If[LessEqual[y, -3.9258428577443835e+52], N[(N[(t - z), $MachinePrecision] * N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.0514448438020326e-28], N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.4
Target	0.4
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if y < -3.92584285774438353e52

   1. Initial program 0.7

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

   2. Simplified 3.0

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

   if -3.92584285774438353e52 < y < 1.0514448438020326e-28

   1. Initial program 2.1

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

   2. Applied egg-rr 2.2

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

   3. Applied egg-rr 0.7

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

   if 1.0514448438020326e-28 < y

   1. Initial program 0.4

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

   2. Taylor expanded in y around 0 20.6

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

   3. Simplified 0.4

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9258428577443835 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{t - a}, x\right)\\ \mathbf{elif}\;y \leq 1.0514448438020326 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(z - t\right), \frac{1}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))))