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

Average Accuracy: 65.8% → 99.9%

Time: 13.0s

Precision: binary64

Cost: 7241

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -19000000 \lor \neg \left(y \leq 350000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x + -1, 1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -19000000.0) (not (<= y 350000.0)))
   (+ (+ x (/ (+ x -1.0) (* y y))) (/ (- 1.0 x) y))
   (fma (/ y (+ y 1.0)) (+ x -1.0) 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 tmp;
	if ((y <= -19000000.0) || !(y <= 350000.0)) {
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	} else {
		tmp = fma((y / (y + 1.0)), (x + -1.0), 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)
	tmp = 0.0
	if ((y <= -19000000.0) || !(y <= 350000.0))
		tmp = Float64(Float64(x + Float64(Float64(x + -1.0) / Float64(y * y))) + Float64(Float64(1.0 - x) / y));
	else
		tmp = fma(Float64(y / Float64(y + 1.0)), Float64(x + -1.0), 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_] := If[Or[LessEqual[y, -19000000.0], N[Not[LessEqual[y, 350000.0]], $MachinePrecision]], N[(N[(x + N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Target

Original	65.8%
Target	99.6%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if y < -1.9e7 or 3.5e5 < y

   1. Initial program 29.4%

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

   2. Simplified 54.3%

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

      [Start] 29.4	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 29.4	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 29.4	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      neg-mul-1 [=>] 29.4	\[ \color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}} + 1 \]
      associate-*l/ [<=] 54.4	\[ -1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)} + 1 \]
      associate-*r* [=>] 54.4	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} + 1 \]
      fma-def [=>] 54.3	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{1 - x}{y + 1}, y, 1\right)} \]
      associate-*r/ [=>] 54.3	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{y + 1}}, y, 1\right) \]
      neg-mul-1 [<=] 54.3	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(1 - x\right)}}{y + 1}, y, 1\right) \]
      neg-sub0 [=>] 54.3	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 54.3	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 54.3	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [<=] 54.3	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 54.3	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Taylor expanded in y around -inf 99.9%

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

   4. Simplified 99.9%

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

      [Start] 99.9	\[ \left(\frac{1}{y} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{y} \]
      associate--l+ [=>] 99.9	\[ \color{blue}{\frac{1}{y} + \left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right)} \]
      +-commutative [=>] 99.9	\[ \color{blue}{\left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right) + \frac{1}{y}} \]
      associate-+l- [=>] 99.9	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \left(\frac{x}{y} - \frac{1}{y}\right)} \]
      +-commutative [=>] 99.9	\[ \color{blue}{\left(x + -1 \cdot \frac{1 - x}{{y}^{2}}\right)} - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      associate-*r/ [=>] 99.9	\[ \left(x + \color{blue}{\frac{-1 \cdot \left(1 - x\right)}{{y}^{2}}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      sub-neg [=>] 99.9	\[ \left(x + \frac{-1 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-in [=>] 99.9	\[ \left(x + \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \left(-x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-commutative [<=] 99.9	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x\right) \cdot -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-neg-in [<=] 99.9	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x \cdot -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-neg-in [=>] 99.9	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{x \cdot \left(--1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      metadata-eval [=>] 99.9	\[ \left(x + \frac{-1 \cdot 1 + x \cdot \color{blue}{1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-in [<=] 99.9	\[ \left(x + \frac{\color{blue}{1 \cdot \left(-1 + x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [<=] 99.9	\[ \left(x + \frac{1 \cdot \color{blue}{\left(x + -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-lft-identity [=>] 99.9	\[ \left(x + \frac{\color{blue}{x + -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [=>] 99.9	\[ \left(x + \frac{\color{blue}{-1 + x}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      unpow2 [=>] 99.9	\[ \left(x + \frac{-1 + x}{\color{blue}{y \cdot y}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      div-sub [<=] 99.9	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \color{blue}{\frac{x - 1}{y}} \]
      sub-neg [=>] 99.9	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 99.9	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{x + \color{blue}{-1}}{y} \]
      +-commutative [=>] 99.9	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{-1 + x}}{y} \]

   if -1.9e7 < y < 3.5e5

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 99.8	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      *-lft-identity [<=] 99.8	\[ \color{blue}{1 \cdot \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} + 1 \]
      associate-/l* [=>] 99.7	\[ 1 \cdot \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\right) + 1 \]
      distribute-neg-frac [=>] 99.7	\[ 1 \cdot \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*r/ [=>] 99.7	\[ \color{blue}{\frac{1 \cdot \left(-\left(1 - x\right)\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*l/ [<=] 99.8	\[ \color{blue}{\frac{1}{\frac{y + 1}{y}} \cdot \left(-\left(1 - x\right)\right)} + 1 \]
      fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{y + 1}{y}}, -\left(1 - x\right), 1\right)} \]
      associate-/l* [<=] 99.8	\[ \mathsf{fma}\left(\color{blue}{\frac{1 \cdot y}{y + 1}}, -\left(1 - x\right), 1\right) \]
      *-lft-identity [=>] 99.8	\[ \mathsf{fma}\left(\frac{\color{blue}{y}}{y + 1}, -\left(1 - x\right), 1\right) \]
      +-commutative [=>] 99.8	\[ \mathsf{fma}\left(\frac{y}{\color{blue}{1 + y}}, -\left(1 - x\right), 1\right) \]
      neg-sub0 [=>] 99.8	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{0 - \left(1 - x\right)}, 1\right) \]
      associate--r- [=>] 99.8	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{\left(0 - 1\right) + x}, 1\right) \]
      metadata-eval [=>] 99.8	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{-1} + x, 1\right) \]
      +-commutative [<=] 99.8	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{x + -1}, 1\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -19000000 \lor \neg \left(y \leq 350000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x + -1, 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;y \leq -19000000 \lor \neg \left(y \leq 350000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -560000000000 \lor \neg \left(y \leq 245000000\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -122000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 245000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4
Accuracy	73.2%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+37}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-75}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.7:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	84.5%
Cost	716

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-78}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	97.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \]

Alternative 7
Accuracy	98.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 0.82:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 8
Accuracy	73.0%
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.35:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	97.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 10
Accuracy	72.7%
Cost	460

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-76}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	73.6%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	38.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023129 
(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))))