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

Average Error: 10.6 → 0.6

Time: 4.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+135}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]

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 (/ (* y (- z t)) (- a t))))
   (if (<= t_1 (- INFINITY))
     (+ x (/ y (/ (- a t) (- z t))))
     (if (<= t_1 2e+135) (+ t_1 x) (fma y (/ (- z t) (- 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 = (y * (z - t)) / (a - t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + (y / ((a - t) / (z - t)));
	} else if (t_1 <= 2e+135) {
		tmp = t_1 + x;
	} else {
		tmp = fma(y, ((z - t) / (a - t)), x);
	}
	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)
	t_1 = Float64(Float64(y * Float64(z - t)) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + Float64(y / Float64(Float64(a - t) / Float64(z - t))));
	elseif (t_1 <= 2e+135)
		tmp = Float64(t_1 + x);
	else
		tmp = fma(y, Float64(Float64(z - t) / Float64(a - t)), x);
	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_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[(y / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+135], N[(t$95$1 + x), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $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	10.6
Target	1.3
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < 1.99999999999999992e135

   1. Initial program 0.2

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

   if 1.99999999999999992e135 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))

   1. Initial program 36.2

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

   2. Simplified 3.4

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \leq -\infty:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))