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

Average Error: 22.0 → 0.6

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(x + \frac{x}{y \cdot y}\right) - \mathsf{fma}\left(x + -1, \frac{1}{y}, {y}^{-2}\right)\\ \mathbf{if}\;y \leq -396.1505766906375:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 435.2526031412426:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x (/ x (* y y))) (fma (+ x -1.0) (/ 1.0 y) (pow y -2.0)))))
   (if (<= y -396.1505766906375)
     t_0
     (if (<= y 435.2526031412426) (fma y (/ x (+ y 1.0)) 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 = (x + (x / (y * y))) - fma((x + -1.0), (1.0 / y), pow(y, -2.0));
	double tmp;
	if (y <= -396.1505766906375) {
		tmp = t_0;
	} else if (y <= 435.2526031412426) {
		tmp = fma(y, (x / (y + 1.0)), 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(x + Float64(x / Float64(y * y))) - fma(Float64(x + -1.0), Float64(1.0 / y), (y ^ -2.0)))
	tmp = 0.0
	if (y <= -396.1505766906375)
		tmp = t_0;
	elseif (y <= 435.2526031412426)
		tmp = fma(y, Float64(x / Float64(y + 1.0)), 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[(x + N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x + -1.0), $MachinePrecision] * N[(1.0 / y), $MachinePrecision] + N[Power[y, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -396.1505766906375], t$95$0, If[LessEqual[y, 435.2526031412426], N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Target

Original	22.0
Target	0.3
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if y < -396.150576690637479 or 435.25260314124262 < y

   1. Initial program 44.8

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

   2. Simplified 28.5

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

   3. Taylor expanded in y around inf 0.2

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

   4. Simplified 0.2

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

   5. Applied egg-rr 0.2

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

   if -396.150576690637479 < y < 435.25260314124262

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   3. Taylor expanded in x around inf 1.0

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -396.1505766906375:\\ \;\;\;\;\left(x + \frac{x}{y \cdot y}\right) - \mathsf{fma}\left(x + -1, \frac{1}{y}, {y}^{-2}\right)\\ \mathbf{elif}\;y \leq 435.2526031412426:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x}{y \cdot y}\right) - \mathsf{fma}\left(x + -1, \frac{1}{y}, {y}^{-2}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))))