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

Average Accuracy: 65.4% → 99.9%

Time: 12.8s

Precision: binary64

Cost: 1225

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -9500000000 \lor \neg \left(y \leq 280000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9500000000.0) (not (<= y 280000.0)))
   (+ (+ x (/ (+ x -1.0) (* y y))) (/ (- 1.0 x) y))
   (+ 1.0 (/ (* y (+ x -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 tmp;
	if ((y <= -9500000000.0) || !(y <= 280000.0)) {
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-9500000000.0d0)) .or. (.not. (y <= 280000.0d0))) then
        tmp = (x + ((x + (-1.0d0)) / (y * y))) + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

↓

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9500000000.0) || !(y <= 280000.0)) {
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	return 1.0 - (((1.0 - x) * y) / (y + 1.0))

↓

def code(x, y):
	tmp = 0
	if (y <= -9500000000.0) or not (y <= 280000.0):
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y)
	else:
		tmp = 1.0 + ((y * (x + -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)
	tmp = 0.0
	if ((y <= -9500000000.0) || !(y <= 280000.0))
		tmp = Float64(Float64(x + Float64(Float64(x + -1.0) / Float64(y * y))) + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9500000000.0) || ~((y <= 280000.0)))
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	else
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	end
	tmp_2 = 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, -9500000000.0], N[Not[LessEqual[y, 280000.0]], $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[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	65.4%
Target	99.7%
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 < -9.5e9 or 2.8e5 < y

   1. Initial program 29.1%

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

   2. Simplified 54.9%

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

      [Start] 29.1	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 29.1	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 29.1	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      associate-/l* [=>] 55.0	\[ \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\right) + 1 \]
      distribute-neg-frac [=>] 55.0	\[ \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-/r/ [=>] 54.9	\[ \color{blue}{\frac{-\left(1 - x\right)}{y + 1} \cdot y} + 1 \]
      fma-def [=>] 54.9	\[ \color{blue}{\mathsf{fma}\left(\frac{-\left(1 - x\right)}{y + 1}, y, 1\right)} \]
      neg-sub0 [=>] 54.9	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 54.9	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 54.9	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [=>] 54.9	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 54.9	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

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

   if -9.5e9 < y < 2.8e5

   1. Initial program 99.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9500000000 \lor \neg \left(y \leq 280000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -35000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 195000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -22000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 195000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 3
Accuracy	98.3%
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 4
Accuracy	98.5%
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 5
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 6
Accuracy	86.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 88:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	74.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00165:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	73.7%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00152:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	38.4%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023151 
(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))))