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

Average Error: 22.6 → 1.3

Time: 13.4s

Precision: binary64

Cost: 8264

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \mathbf{elif}\;t_0 \leq 1.1:\\ \;\;\;\;x + \left(\frac{1 + \frac{1}{y \cdot y}}{y} + \frac{-1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x + -1, 1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (<= t_0 5e-24)
     (+ 1.0 (* y (/ (+ x -1.0) (+ 1.0 y))))
     (if (<= t_0 1.1)
       (+ x (+ (/ (+ 1.0 (/ 1.0 (* y y))) y) (/ -1.0 (* y y))))
       (fma (/ y (+ 1.0 y)) (+ 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 t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if (t_0 <= 5e-24) {
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)));
	} else if (t_0 <= 1.1) {
		tmp = x + (((1.0 + (1.0 / (y * y))) / y) + (-1.0 / (y * y)));
	} else {
		tmp = fma((y / (1.0 + y)), (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)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if (t_0 <= 5e-24)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(1.0 + y))));
	elseif (t_0 <= 1.1)
		tmp = Float64(x + Float64(Float64(Float64(1.0 + Float64(1.0 / Float64(y * y))) / y) + Float64(-1.0 / Float64(y * y))));
	else
		tmp = fma(Float64(y / Float64(1.0 + y)), 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_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-24], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.1], N[(x + N[(N[(N[(1.0 + N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Target

Original	22.6
Target	0.2
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 4.9999999999999998e-24

   1. Initial program 7.5

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

   2. Simplified 0.0

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

      [Start] 7.5	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      remove-double-neg [<=] 7.5	\[ 1 - \color{blue}{\left(-\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)\right)} \]
      neg-mul-1 [=>] 7.5	\[ 1 - \left(-\color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}}\right) \]
      associate-*l/ [<=] 0.0	\[ 1 - \left(--1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)}\right) \]
      associate-*r* [=>] 0.0	\[ 1 - \left(-\color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y}\right) \]
      distribute-lft-neg-in [=>] 0.0	\[ 1 - \color{blue}{\left(--1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} \]
      distribute-lft-neg-in [=>] 0.0	\[ 1 - \color{blue}{\left(\left(--1\right) \cdot \frac{1 - x}{y + 1}\right)} \cdot y \]
      metadata-eval [=>] 0.0	\[ 1 - \left(\color{blue}{1} \cdot \frac{1 - x}{y + 1}\right) \cdot y \]
      *-lft-identity [=>] 0.0	\[ 1 - \color{blue}{\frac{1 - x}{y + 1}} \cdot y \]
      +-commutative [=>] 0.0	\[ 1 - \frac{1 - x}{\color{blue}{1 + y}} \cdot y \]

   if 4.9999999999999998e-24 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.1000000000000001

   1. Initial program 53.7

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

   2. Simplified 53.7

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

      [Start] 53.7	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 53.7	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 53.7	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      *-lft-identity [<=] 53.7	\[ \color{blue}{1 \cdot \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} + 1 \]
      associate-/l* [=>] 53.8	\[ 1 \cdot \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\right) + 1 \]
      distribute-neg-frac [=>] 53.8	\[ 1 \cdot \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*r/ [=>] 53.8	\[ \color{blue}{\frac{1 \cdot \left(-\left(1 - x\right)\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*l/ [<=] 53.8	\[ \color{blue}{\frac{1}{\frac{y + 1}{y}} \cdot \left(-\left(1 - x\right)\right)} + 1 \]
      fma-def [=>] 53.8	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{y + 1}{y}}, -\left(1 - x\right), 1\right)} \]
      associate-/l* [<=] 53.7	\[ \mathsf{fma}\left(\color{blue}{\frac{1 \cdot y}{y + 1}}, -\left(1 - x\right), 1\right) \]
      *-lft-identity [=>] 53.7	\[ \mathsf{fma}\left(\frac{\color{blue}{y}}{y + 1}, -\left(1 - x\right), 1\right) \]
      +-commutative [=>] 53.7	\[ \mathsf{fma}\left(\frac{y}{\color{blue}{1 + y}}, -\left(1 - x\right), 1\right) \]
      neg-sub0 [=>] 53.7	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{0 - \left(1 - x\right)}, 1\right) \]
      associate--r- [=>] 53.7	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{\left(0 - 1\right) + x}, 1\right) \]
      metadata-eval [=>] 53.7	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{-1} + x, 1\right) \]
      +-commutative [<=] 53.7	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{x + -1}, 1\right) \]

   3. Taylor expanded in y around inf 4.5

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

   4. Simplified 4.5

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

      [Start] 4.5	\[ \left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right)\right) - \frac{1}{{y}^{2}} \]
      sub-neg [=>] 4.5	\[ \color{blue}{\left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right)\right) + \left(-\frac{1}{{y}^{2}}\right)} \]
      +-commutative [=>] 4.5	\[ \color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \frac{x}{{y}^{2}}\right)} + \left(-\frac{1}{{y}^{2}}\right) \]
      associate-+l+ [=>] 4.5	\[ \color{blue}{\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} + \left(-\frac{1}{{y}^{2}}\right)\right)} \]
      associate-+r+ [=>] 4.5	\[ \color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + x\right)} + \left(\frac{x}{{y}^{2}} + \left(-\frac{1}{{y}^{2}}\right)\right) \]
      +-commutative [=>] 4.5	\[ \color{blue}{\left(x + \left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right)\right)} + \left(\frac{x}{{y}^{2}} + \left(-\frac{1}{{y}^{2}}\right)\right) \]
      associate-+l+ [=>] 4.5	\[ \color{blue}{x + \left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \left(\frac{x}{{y}^{2}} + \left(-\frac{1}{{y}^{2}}\right)\right)\right)} \]

   5. Taylor expanded in x around 0 4.6

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

   6. Simplified 4.6

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

      [Start] 4.6	\[ x + \left(\frac{1 + \frac{1}{{y}^{2}}}{y} + \frac{x + -1}{y \cdot y}\right) \]
      unpow2 [=>] 4.6	\[ x + \left(\frac{1 + \frac{1}{\color{blue}{y \cdot y}}}{y} + \frac{x + -1}{y \cdot y}\right) \]

   7. Taylor expanded in x around 0 4.6

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

   8. Simplified 4.6

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

      [Start] 4.6	\[ x + \left(\frac{1 + \frac{1}{y \cdot y}}{y} + \frac{-1}{{y}^{2}}\right) \]
      unpow2 [=>] 4.6	\[ x + \left(\frac{1 + \frac{1}{y \cdot y}}{y} + \frac{-1}{\color{blue}{y \cdot y}}\right) \]

   if 1.1000000000000001 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 21.8

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

   2. Simplified 0.0

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

      [Start] 21.8	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 21.8	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 21.8	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      *-lft-identity [<=] 21.8	\[ \color{blue}{1 \cdot \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} + 1 \]
      associate-/l* [=>] 0.1	\[ 1 \cdot \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\right) + 1 \]
      distribute-neg-frac [=>] 0.1	\[ 1 \cdot \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*r/ [=>] 0.1	\[ \color{blue}{\frac{1 \cdot \left(-\left(1 - x\right)\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*l/ [<=] 0.1	\[ \color{blue}{\frac{1}{\frac{y + 1}{y}} \cdot \left(-\left(1 - x\right)\right)} + 1 \]
      fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{y + 1}{y}}, -\left(1 - x\right), 1\right)} \]
      associate-/l* [<=] 0.0	\[ \mathsf{fma}\left(\color{blue}{\frac{1 \cdot y}{y + 1}}, -\left(1 - x\right), 1\right) \]
      *-lft-identity [=>] 0.0	\[ \mathsf{fma}\left(\frac{\color{blue}{y}}{y + 1}, -\left(1 - x\right), 1\right) \]
      +-commutative [=>] 0.0	\[ \mathsf{fma}\left(\frac{y}{\color{blue}{1 + y}}, -\left(1 - x\right), 1\right) \]
      neg-sub0 [=>] 0.0	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{0 - \left(1 - x\right)}, 1\right) \]
      associate--r- [=>] 0.0	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{\left(0 - 1\right) + x}, 1\right) \]
      metadata-eval [=>] 0.0	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{-1} + x, 1\right) \]
      +-commutative [<=] 0.0	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{x + -1}, 1\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.1:\\ \;\;\;\;x + \left(\frac{1 + \frac{1}{y \cdot y}}{y} + \frac{-1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x + -1, 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1352

\[\begin{array}{l} \mathbf{if}\;y \leq -232000:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 3100000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 + \frac{1}{y \cdot y}}{y} + \frac{-1}{y \cdot y}\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1225

\[\begin{array}{l} \mathbf{if}\;y \leq -270000 \lor \neg \left(y \leq 235000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -88000000000 \lor \neg \left(y \leq 20500000000\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -16000000000 \lor \neg \left(y \leq 30000000000\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \end{array} \]

Alternative 5
Error	18.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	1.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.86\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot y - y\right)\\ \end{array} \]

Alternative 7
Error	8.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.75 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 8
Error	1.3
Cost	585

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

Alternative 9
Error	16.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	16.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	39.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023066 
(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))))