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

Average Error: 22.2 → 0.5

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ t_1 := \mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;\frac{1}{y} + \left(\left(x + \left({y}^{-3} + x \cdot {y}^{-2}\right)\right) - \left(\frac{x}{y} + \mathsf{fma}\left(x, {y}^{-3}, {y}^{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 x) y) (+ 1.0 y)))
        (t_1 (fma (- 1.0 x) (/ y (- -1.0 y)) 1.0)))
   (if (<= t_0 5e-7)
     t_1
     (if (<= t_0 2.0)
       (+
        (/ 1.0 y)
        (-
         (+ x (+ (pow y -3.0) (* x (pow y -2.0))))
         (+ (/ x y) (fma x (pow y -3.0) (pow y -2.0)))))
       t_1))))

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 - x) * y) / (1.0 + y);
	double t_1 = fma((1.0 - x), (y / (-1.0 - y)), 1.0);
	double tmp;
	if (t_0 <= 5e-7) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = (1.0 / y) + ((x + (pow(y, -3.0) + (x * pow(y, -2.0)))) - ((x / y) + fma(x, pow(y, -3.0), pow(y, -2.0))));
	} else {
		tmp = t_1;
	}
	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 - x) * y) / Float64(1.0 + y))
	t_1 = fma(Float64(1.0 - x), Float64(y / Float64(-1.0 - y)), 1.0)
	tmp = 0.0
	if (t_0 <= 5e-7)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(1.0 / y) + Float64(Float64(x + Float64((y ^ -3.0) + Float64(x * (y ^ -2.0)))) - Float64(Float64(x / y) + fma(x, (y ^ -3.0), (y ^ -2.0)))));
	else
		tmp = t_1;
	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 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-7], t$95$1, If[LessEqual[t$95$0, 2.0], N[(N[(1.0 / y), $MachinePrecision] + N[(N[(x + N[(N[Power[y, -3.0], $MachinePrecision] + N[(x * N[Power[y, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] + N[(x * N[Power[y, -3.0], $MachinePrecision] + N[Power[y, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Target

Original	22.2
Target	0.2
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 4.99999999999999977e-7 or 2 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 10.7

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

   2. Simplified 0.0

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

   if 4.99999999999999977e-7 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 2

   1. Initial program 56.1

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

   2. Simplified 56.1

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

   3. Taylor expanded in y around inf 1.8

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

   4. Applied egg-rr 1.8

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

   5. Applied egg-rr 1.8

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 2:\\ \;\;\;\;\frac{1}{y} + \left(\left(x + \left({y}^{-3} + x \cdot {y}^{-2}\right)\right) - \left(\frac{x}{y} + \mathsf{fma}\left(x, {y}^{-3}, {y}^{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))