Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Average Error: 23.1 → 0.1

Time: 3.3s

Precision: binary64

Math

\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{y} + \mathsf{fma}\left(1, \mathsf{fma}\left(x, {y}^{-2}, x\right), {y}^{-3}\right)\right) + \left(\left(\frac{-1}{{y}^{2}} - \frac{x}{{y}^{3}}\right) - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1490796221.341475:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 11397.938900905672:\\ \;\;\;\;\left(1 + \frac{y \cdot x}{y + 1}\right) - \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (+ (/ 1.0 y) (fma 1.0 (fma x (pow y -2.0) x) (pow y -3.0)))
          (- (- (/ -1.0 (pow y 2.0)) (/ x (pow y 3.0))) (/ x y)))))
   (if (<= y -1490796221.341475)
     t_0
     (if (<= y 11397.938900905672)
       (- (+ 1.0 (/ (* y x) (+ y 1.0))) (/ y (+ y 1.0)))
       t_0))))

C

double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

↓

double code(double x, double y) {
	double t_0 = ((1.0 / y) + fma(1.0, fma(x, pow(y, -2.0), x), pow(y, -3.0))) + (((-1.0 / pow(y, 2.0)) - (x / pow(y, 3.0))) - (x / y));
	double tmp;
	if (y <= -1490796221.341475) {
		tmp = t_0;
	} else if (y <= 11397.938900905672) {
		tmp = (1.0 + ((y * x) / (y + 1.0))) - (y / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	return Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)))
end

↓

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 / y) + fma(1.0, fma(x, (y ^ -2.0), x), (y ^ -3.0))) + Float64(Float64(Float64(-1.0 / (y ^ 2.0)) - Float64(x / (y ^ 3.0))) - Float64(x / y)))
	tmp = 0.0
	if (y <= -1490796221.341475)
		tmp = t_0;
	elseif (y <= 11397.938900905672)
		tmp = Float64(Float64(1.0 + Float64(Float64(y * x) / Float64(y + 1.0))) - Float64(y / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 / y), $MachinePrecision] + N[(1.0 * N[(x * N[Power[y, -2.0], $MachinePrecision] + x), $MachinePrecision] + N[Power[y, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1490796221.341475], t$95$0, If[LessEqual[y, 11397.938900905672], N[(N[(1.0 + N[(N[(y * x), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Target

Original	23.1
Target	0.2
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -1490796221.341475 or 11397.938900905672 < y

   1. Initial program 46.5

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Simplified 29.2

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

   3. Taylor expanded in y around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \left(\frac{x}{{y}^{2}} + x\right)\right)\right) - \left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)} \]

   4. Applied egg-rr 0.0

      \[\leadsto \left(\frac{1}{y} + \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(x, {y}^{-2}, x\right), {y}^{-3}\right)}\right) - \left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right) \]

   if -1490796221.341475 < y < 11397.938900905672

   1. Initial program 0.1

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Simplified 0.1

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

   3. Taylor expanded in x around 0 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1490796221.341475:\\ \;\;\;\;\left(\frac{1}{y} + \mathsf{fma}\left(1, \mathsf{fma}\left(x, {y}^{-2}, x\right), {y}^{-3}\right)\right) + \left(\left(\frac{-1}{{y}^{2}} - \frac{x}{{y}^{3}}\right) - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 11397.938900905672:\\ \;\;\;\;\left(1 + \frac{y \cdot x}{y + 1}\right) - \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{y} + \mathsf{fma}\left(1, \mathsf{fma}\left(x, {y}^{-2}, x\right), {y}^{-3}\right)\right) + \left(\left(\frac{-1}{{y}^{2}} - \frac{x}{{y}^{3}}\right) - \frac{x}{y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022151 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))