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

Average Accuracy: 64.2% → 99.8%

Time: 19.1s

Precision: binary64

Cost: 7240

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -4300000000:\\ \;\;\;\;x + t_0\\ \mathbf{elif}\;y \leq 62000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + t_0\\ \end{array} \]

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 x) y)))
   (if (<= y -4300000000.0)
     (+ x t_0)
     (if (<= y 62000000.0)
       (fma (/ y (+ y 1.0)) (+ x -1.0) 1.0)
       (+ (+ x (/ (+ x -1.0) (* y y))) 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 - x) / y;
	double tmp;
	if (y <= -4300000000.0) {
		tmp = x + t_0;
	} else if (y <= 62000000.0) {
		tmp = fma((y / (y + 1.0)), (x + -1.0), 1.0);
	} else {
		tmp = (x + ((x + -1.0) / (y * y))) + 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(1.0 - x) / y)
	tmp = 0.0
	if (y <= -4300000000.0)
		tmp = Float64(x + t_0);
	elseif (y <= 62000000.0)
		tmp = fma(Float64(y / Float64(y + 1.0)), Float64(x + -1.0), 1.0);
	else
		tmp = Float64(Float64(x + Float64(Float64(x + -1.0) / Float64(y * y))) + 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[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -4300000000.0], N[(x + t$95$0), $MachinePrecision], If[LessEqual[y, 62000000.0], N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x + N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Target

Original	64.2%
Target	99.7%
Herbie	99.8%

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

Derivation

1. Split input into 3 regimes

2. if y < -4.3e9

   1. Initial program 28.6%

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

   2. Simplified 54.1%

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

      [Start] 28.6	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 28.6	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      distribute-neg-frac [=>] 28.6	\[ 1 + \color{blue}{\frac{-\left(1 - x\right) \cdot y}{y + 1}} \]
      neg-mul-1 [=>] 28.6	\[ 1 + \frac{\color{blue}{-1 \cdot \left(\left(1 - x\right) \cdot y\right)}}{y + 1} \]
      associate-*l/ [<=] 28.5	\[ 1 + \color{blue}{\frac{-1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)} \]
      metadata-eval [<=] 28.5	\[ 1 + \frac{\color{blue}{1 \cdot -1}}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      associate-*l/ [<=] 28.5	\[ 1 + \color{blue}{\left(\frac{1}{y + 1} \cdot -1\right)} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      associate-/r/ [<=] 28.5	\[ 1 + \color{blue}{\frac{1}{\frac{y + 1}{-1}}} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      metadata-eval [<=] 28.5	\[ 1 + \frac{\color{blue}{--1}}{\frac{y + 1}{-1}} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      distribute-neg-frac [<=] 28.5	\[ 1 + \color{blue}{\left(-\frac{-1}{\frac{y + 1}{-1}}\right)} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      cancel-sign-sub-inv [<=] 28.5	\[ \color{blue}{1 - \frac{-1}{\frac{y + 1}{-1}} \cdot \left(\left(1 - x\right) \cdot y\right)} \]
      associate-/r/ [<=] 28.5	\[ 1 - \color{blue}{\frac{-1}{\frac{\frac{y + 1}{-1}}{\left(1 - x\right) \cdot y}}} \]
      associate-/r* [<=] 28.5	\[ 1 - \frac{-1}{\color{blue}{\frac{y + 1}{-1 \cdot \left(\left(1 - x\right) \cdot y\right)}}} \]
      neg-mul-1 [<=] 28.5	\[ 1 - \frac{-1}{\frac{y + 1}{\color{blue}{-\left(1 - x\right) \cdot y}}} \]
      associate-/r/ [=>] 28.5	\[ 1 - \color{blue}{\frac{-1}{y + 1} \cdot \left(-\left(1 - x\right) \cdot y\right)} \]
      distribute-rgt-neg-in [<=] 28.5	\[ 1 - \color{blue}{\left(-\frac{-1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\right)} \]
      associate-/r/ [<=] 28.5	\[ 1 - \left(-\color{blue}{\frac{-1}{\frac{y + 1}{\left(1 - x\right) \cdot y}}}\right) \]
      distribute-neg-frac [=>] 28.5	\[ 1 - \color{blue}{\frac{--1}{\frac{y + 1}{\left(1 - x\right) \cdot y}}} \]
      metadata-eval [=>] 28.5	\[ 1 - \frac{\color{blue}{1}}{\frac{y + 1}{\left(1 - x\right) \cdot y}} \]
      associate-/r/ [=>] 28.5	\[ 1 - \color{blue}{\frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)} \]

   3. Taylor expanded in y around inf 99.8%

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

   4. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{x + -1}{y}} \]
      Proof

      [Start] 99.8	\[ \left(\frac{1}{y} + x\right) - \frac{x}{y} \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(x + \frac{1}{y}\right)} - \frac{x}{y} \]
      associate--l+ [=>] 99.8	\[ \color{blue}{x + \left(\frac{1}{y} - \frac{x}{y}\right)} \]
      div-sub [<=] 99.8	\[ x + \color{blue}{\frac{1 - x}{y}} \]
      sub-neg [=>] 99.8	\[ x + \frac{\color{blue}{1 + \left(-x\right)}}{y} \]
      +-commutative [=>] 99.8	\[ x + \frac{\color{blue}{\left(-x\right) + 1}}{y} \]
      neg-sub0 [=>] 99.8	\[ x + \frac{\color{blue}{\left(0 - x\right)} + 1}{y} \]
      associate-+l- [=>] 99.8	\[ x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]
      neg-sub0 [<=] 99.8	\[ x + \frac{\color{blue}{-\left(x - 1\right)}}{y} \]
      mul-1-neg [<=] 99.8	\[ x + \frac{\color{blue}{-1 \cdot \left(x - 1\right)}}{y} \]
      associate-*r/ [<=] 99.8	\[ x + \color{blue}{-1 \cdot \frac{x - 1}{y}} \]
      mul-1-neg [=>] 99.8	\[ x + \color{blue}{\left(-\frac{x - 1}{y}\right)} \]
      unsub-neg [=>] 99.8	\[ \color{blue}{x - \frac{x - 1}{y}} \]
      sub-neg [=>] 99.8	\[ x - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 99.8	\[ x - \frac{x + \color{blue}{-1}}{y} \]

   if -4.3e9 < y < 6.2e7

   1. Initial program 99.7%

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

   2. Simplified 99.8%

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

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

   if 6.2e7 < y

   1. Initial program 27.6%

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

   2. Simplified 52.8%

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

      [Start] 27.6	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 27.6	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 27.6	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      associate-/l* [=>] 52.9	\[ \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\right) + 1 \]
      distribute-neg-frac [=>] 52.9	\[ \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-/r/ [=>] 52.9	\[ \color{blue}{\frac{-\left(1 - x\right)}{y + 1} \cdot y} + 1 \]
      fma-def [=>] 52.8	\[ \color{blue}{\mathsf{fma}\left(\frac{-\left(1 - x\right)}{y + 1}, y, 1\right)} \]
      neg-sub0 [=>] 52.8	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 52.8	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 52.8	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [=>] 52.8	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 52.8	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Taylor expanded in y around -inf 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ \left(\frac{1}{y} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{y} \]
      +-commutative [=>] 100.0	\[ \color{blue}{\left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) + \frac{1}{y}\right)} - \frac{x}{y} \]
      associate--l+ [=>] 100.0	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) + \left(\frac{1}{y} - \frac{x}{y}\right)} \]
      +-commutative [=>] 100.0	\[ \color{blue}{\left(x + -1 \cdot \frac{1 - x}{{y}^{2}}\right)} + \left(\frac{1}{y} - \frac{x}{y}\right) \]
      mul-1-neg [=>] 100.0	\[ \left(x + \color{blue}{\left(-\frac{1 - x}{{y}^{2}}\right)}\right) + \left(\frac{1}{y} - \frac{x}{y}\right) \]
      unsub-neg [=>] 100.0	\[ \color{blue}{\left(x - \frac{1 - x}{{y}^{2}}\right)} + \left(\frac{1}{y} - \frac{x}{y}\right) \]
      unpow2 [=>] 100.0	\[ \left(x - \frac{1 - x}{\color{blue}{y \cdot y}}\right) + \left(\frac{1}{y} - \frac{x}{y}\right) \]
      div-sub [<=] 100.0	\[ \left(x - \frac{1 - x}{y \cdot y}\right) + \color{blue}{\frac{1 - x}{y}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4300000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 62000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1224

\[\begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -156000000:\\ \;\;\;\;x + t_0\\ \mathbf{elif}\;y \leq 62000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + t_0\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -156000000 \lor \neg \left(y \leq 175000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 3
Accuracy	86.4%
Cost	716

\[\begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-30}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.43:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	98.3%
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 5
Accuracy	98.6%
Cost	713

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

Alternative 6
Accuracy	74.4%
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{-33}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.78:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	98.0%
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 8
Accuracy	74.3%
Cost	460

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-32}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 330:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	74.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 330:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	38.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023152 
(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))))