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

Average Error: 22.4 → 0.0

Time: 9.2s

Precision: binary64

Cost: 7241

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-1 + x}{y \cdot y}\\ \mathbf{if}\;y \leq -13200 \lor \neg \left(y \leq 14200\right):\\ \;\;\;\;\left(t_0 + \left(x - t_0 \cdot \frac{1}{y}\right)\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1 + x}{y + 1}, y, 1\right)\\ \end{array} \]

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 y))))
   (if (or (<= y -13200.0) (not (<= y 14200.0)))
     (+ (+ t_0 (- x (* t_0 (/ 1.0 y)))) (/ (- 1.0 x) y))
     (fma (/ (+ -1.0 x) (+ 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 * y);
	double tmp;
	if ((y <= -13200.0) || !(y <= 14200.0)) {
		tmp = (t_0 + (x - (t_0 * (1.0 / y)))) + ((1.0 - x) / y);
	} else {
		tmp = fma(((-1.0 + x) / (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(-1.0 + x) / Float64(y * y))
	tmp = 0.0
	if ((y <= -13200.0) || !(y <= 14200.0))
		tmp = Float64(Float64(t_0 + Float64(x - Float64(t_0 * Float64(1.0 / y)))) + Float64(Float64(1.0 - x) / y));
	else
		tmp = fma(Float64(Float64(-1.0 + x) / Float64(y + 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[(-1.0 + x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -13200.0], N[Not[LessEqual[y, 14200.0]], $MachinePrecision]], N[(N[(t$95$0 + N[(x - N[(t$95$0 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 + x), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Target

Original	22.4
Target	0.2
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if y < -13200 or 14200 < y

   1. Initial program 45.4

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

   2. Simplified 29.3

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

      [Start] 45.4	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 45.4	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 45.4	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      neg-mul-1 [=>] 45.4	\[ \color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}} + 1 \]
      associate-*l/ [<=] 29.3	\[ -1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)} + 1 \]
      associate-*r* [=>] 29.3	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} + 1 \]
      fma-def [=>] 29.3	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{1 - x}{y + 1}, y, 1\right)} \]
      associate-*r/ [=>] 29.3	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{y + 1}}, y, 1\right) \]
      neg-mul-1 [<=] 29.3	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(1 - x\right)}}{y + 1}, y, 1\right) \]
      neg-sub0 [=>] 29.3	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 29.3	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 29.3	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [<=] 29.3	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 29.3	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Taylor expanded in y around -inf 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \frac{x}{y}\right) \]
      associate--l+ [=>] 0.0	\[ \color{blue}{\frac{1}{y} + \left(\left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \left(\frac{x}{{y}^{3}} + \frac{x}{y}\right)\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \left(\frac{x}{{y}^{3}} + \frac{x}{y}\right)\right) + \frac{1}{y}} \]
      associate--r+ [=>] 0.0	\[ \color{blue}{\left(\left(\left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{{y}^{3}}\right) - \frac{x}{y}\right)} + \frac{1}{y} \]
      associate-+l- [=>] 0.0	\[ \color{blue}{\left(\left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{{y}^{3}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right)} \]

   5. Applied egg-rr 0.0

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

   if -13200 < y < 14200

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 0.0	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      neg-mul-1 [=>] 0.0	\[ \color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}} + 1 \]
      associate-*l/ [<=] 0.0	\[ -1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)} + 1 \]
      associate-*r* [=>] 0.0	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} + 1 \]
      fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{1 - x}{y + 1}, y, 1\right)} \]
      associate-*r/ [=>] 0.0	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{y + 1}}, y, 1\right) \]
      neg-mul-1 [<=] 0.0	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(1 - x\right)}}{y + 1}, y, 1\right) \]
      neg-sub0 [=>] 0.0	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 0.0	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 0.0	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [<=] 0.0	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 0.0	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13200 \lor \neg \left(y \leq 14200\right):\\ \;\;\;\;\left(\frac{-1 + x}{y \cdot y} + \left(x - \frac{-1 + x}{y \cdot y} \cdot \frac{1}{y}\right)\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1 + x}{y + 1}, y, 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	1993

\[\begin{array}{l} t_0 := \frac{-1 + x}{y \cdot y}\\ \mathbf{if}\;y \leq -13200 \lor \neg \left(y \leq 11400\right):\\ \;\;\;\;\left(t_0 + \left(x - t_0 \cdot \frac{1}{y}\right)\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-1 + x}{y + 1}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1092

\[\begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -300000:\\ \;\;\;\;\left(x + \frac{-1 + x}{y \cdot y}\right) + t_0\\ \mathbf{elif}\;y \leq 122000000:\\ \;\;\;\;1 + y \cdot \frac{-1 + x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -8100000000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 122000000:\\ \;\;\;\;1 + y \cdot \frac{-1 + x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 4
Error	1.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.8\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \]

Alternative 5
Error	1.0
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
Error	16.4
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 0.395:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	8.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.059\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 8
Error	1.3
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 9
Error	16.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.49:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	16.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	39.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023237 
(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))))