Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 1.9 → 1.5

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.802344879117195 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{t}, x\right)\\ \mathbf{elif}\;t \leq 7.032798435538504 \cdot 10^{+36}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.802344879117195e+58)
   (fma z (/ (- y x) t) x)
   (if (<= t 7.032798435538504e+36)
     (+ x (/ (* z (- y x)) t))
     (fma (/ z t) (- y x) x))))

C

double code(double x, double y, double z, double t) {
	return x + ((y - x) * (z / t));
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.802344879117195e+58) {
		tmp = fma(z, ((y - x) / t), x);
	} else if (t <= 7.032798435538504e+36) {
		tmp = x + ((z * (y - x)) / t);
	} else {
		tmp = fma((z / t), (y - x), x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.802344879117195e+58)
		tmp = fma(z, Float64(Float64(y - x) / t), x);
	elseif (t <= 7.032798435538504e+36)
		tmp = Float64(x + Float64(Float64(z * Float64(y - x)) / t));
	else
		tmp = fma(Float64(z / t), Float64(y - x), x);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[t, -1.802344879117195e+58], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 7.032798435538504e+36], N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.9
Target	2.1
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} < -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if t < -1.802344879117195e58

   1. Initial program 1.1

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

   2. Taylor expanded in x around 0 10.6

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

   3. Simplified 1.1

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

   if -1.802344879117195e58 < t < 7.0327984355385045e36

   1. Initial program 3.0

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

   2. Taylor expanded in z around 0 2.0

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

   if 7.0327984355385045e36 < t

   1. Initial program 1.0

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

   2. Applied egg-rr 1.0

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.802344879117195 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{t}, x\right)\\ \mathbf{elif}\;t \leq 7.032798435538504 \cdot 10^{+36}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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