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

Average Error: 23.1 → 1.1

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t_0 \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, -\frac{y}{1 + y}, 1\right)\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;\left(\frac{1}{y} + \left(x + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{x}{y} + \sqrt[3]{{y}^{-6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{-1 - y}\right)\right), 1\right)\\ \end{array} \]

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))))
   (if (<= t_0 1e-19)
     (fma (- 1.0 x) (- (/ y (+ 1.0 y))) 1.0)
     (if (<= t_0 2.0)
       (-
        (+ (/ 1.0 y) (+ x (/ x (pow y 2.0))))
        (+ (/ x y) (cbrt (pow y -6.0))))
       (fma (- 1.0 x) (log1p (expm1 (/ y (- -1.0 y)))) 1.0)))))

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 - x) * y) / (1.0 + y);
	double tmp;
	if (t_0 <= 1e-19) {
		tmp = fma((1.0 - x), -(y / (1.0 + y)), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = ((1.0 / y) + (x + (x / pow(y, 2.0)))) - ((x / y) + cbrt(pow(y, -6.0)));
	} else {
		tmp = fma((1.0 - x), log1p(expm1((y / (-1.0 - y)))), 1.0);
	}
	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 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= 1e-19)
		tmp = fma(Float64(1.0 - x), Float64(-Float64(y / Float64(1.0 + y))), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(Float64(1.0 / y) + Float64(x + Float64(x / (y ^ 2.0)))) - Float64(Float64(x / y) + cbrt((y ^ -6.0))));
	else
		tmp = fma(Float64(1.0 - x), log1p(expm1(Float64(y / Float64(-1.0 - y)))), 1.0);
	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 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-19], N[(N[(1.0 - x), $MachinePrecision] * (-N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]) + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(1.0 / y), $MachinePrecision] + N[(x + N[(x / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] + N[Power[N[Power[y, -6.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * N[Log[1 + N[(Exp[N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Target

Original	23.1
Target	0.2
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 9.9999999999999998e-20

   1. Initial program 7.7

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

   2. Simplified 0.0

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

   3. Applied egg-rr 0.0

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

   if 9.9999999999999998e-20 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 2

   1. Initial program 54.3

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

   2. Simplified 54.2

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

   3. Taylor expanded in y around inf 4.1

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

   4. Applied egg-rr 4.1

      \[\leadsto \left(\frac{1}{y} + \left(\frac{x}{{y}^{2}} + x\right)\right) - \left(\frac{x}{y} + \color{blue}{\sqrt[3]{{y}^{-6}}}\right) \]

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

   1. Initial program 21.5

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

   2. Simplified 0.0

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

   3. Applied egg-rr 0.0

      \[\leadsto \mathsf{fma}\left(1 - x, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{-1 - y}\right)\right)}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, -\frac{y}{1 + y}, 1\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 2:\\ \;\;\;\;\left(\frac{1}{y} + \left(x + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{x}{y} + \sqrt[3]{{y}^{-6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{-1 - y}\right)\right), 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(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))))