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

Average Accuracy: 63.8% → 99.8%

Time: 15.2s

Precision: binary64

Cost: 1860

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-1}{y \cdot y}\\ \mathbf{if}\;y \leq -12000:\\ \;\;\;\;\frac{x + -1}{y} \cdot \left(-1 + t_0\right) + \left(x + \left(t_0 + \frac{x}{y \cdot y}\right)\right)\\ \mathbf{elif}\;y \leq 6100000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

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 (* y y))))
   (if (<= y -12000.0)
     (+ (* (/ (+ x -1.0) y) (+ -1.0 t_0)) (+ x (+ t_0 (/ x (* y y)))))
     (if (<= y 6100000000.0)
       (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
       (+ x (/ 1.0 y))))))

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 / (y * y);
	double tmp;
	if (y <= -12000.0) {
		tmp = (((x + -1.0) / y) * (-1.0 + t_0)) + (x + (t_0 + (x / (y * y))));
	} else if (y <= 6100000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = x + (1.0 / y);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / (y * y)
    if (y <= (-12000.0d0)) then
        tmp = (((x + (-1.0d0)) / y) * ((-1.0d0) + t_0)) + (x + (t_0 + (x / (y * y))))
    else if (y <= 6100000000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (y + 1.0d0))
    else
        tmp = x + (1.0d0 / y)
    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 t_0 = -1.0 / (y * y);
	double tmp;
	if (y <= -12000.0) {
		tmp = (((x + -1.0) / y) * (-1.0 + t_0)) + (x + (t_0 + (x / (y * y))));
	} else if (y <= 6100000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = -1.0 / (y * y)
	tmp = 0
	if y <= -12000.0:
		tmp = (((x + -1.0) / y) * (-1.0 + t_0)) + (x + (t_0 + (x / (y * y))))
	elif y <= 6100000000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0))
	else:
		tmp = x + (1.0 / y)
	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(-1.0 / Float64(y * y))
	tmp = 0.0
	if (y <= -12000.0)
		tmp = Float64(Float64(Float64(Float64(x + -1.0) / y) * Float64(-1.0 + t_0)) + Float64(x + Float64(t_0 + Float64(x / Float64(y * y)))));
	elseif (y <= 6100000000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0)));
	else
		tmp = Float64(x + Float64(1.0 / y));
	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)
	t_0 = -1.0 / (y * y);
	tmp = 0.0;
	if (y <= -12000.0)
		tmp = (((x + -1.0) / y) * (-1.0 + t_0)) + (x + (t_0 + (x / (y * y))));
	elseif (y <= 6100000000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	else
		tmp = x + (1.0 / y);
	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_] := Block[{t$95$0 = N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -12000.0], N[(N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(x + N[(t$95$0 + N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6100000000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	63.8%
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 < -12000

   1. Initial program 29.0%

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

   2. Simplified 55.4%

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

      [Start] 29.0	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 29.0	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 29.0	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      *-lft-identity [<=] 29.0	\[ \color{blue}{1 \cdot \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} + 1 \]
      associate-/l* [=>] 55.4	\[ 1 \cdot \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\right) + 1 \]
      distribute-neg-frac [=>] 55.4	\[ 1 \cdot \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*r/ [=>] 55.4	\[ \color{blue}{\frac{1 \cdot \left(-\left(1 - x\right)\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*l/ [<=] 55.3	\[ \color{blue}{\frac{1}{\frac{y + 1}{y}} \cdot \left(-\left(1 - x\right)\right)} + 1 \]
      fma-def [=>] 55.3	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{y + 1}{y}}, -\left(1 - x\right), 1\right)} \]
      associate-/l* [<=] 55.4	\[ \mathsf{fma}\left(\color{blue}{\frac{1 \cdot y}{y + 1}}, -\left(1 - x\right), 1\right) \]
      *-lft-identity [=>] 55.4	\[ \mathsf{fma}\left(\frac{\color{blue}{y}}{y + 1}, -\left(1 - x\right), 1\right) \]
      +-commutative [=>] 55.4	\[ \mathsf{fma}\left(\frac{y}{\color{blue}{1 + y}}, -\left(1 - x\right), 1\right) \]
      neg-sub0 [=>] 55.4	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{0 - \left(1 - x\right)}, 1\right) \]
      associate--r- [=>] 55.4	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{\left(0 - 1\right) + x}, 1\right) \]
      metadata-eval [=>] 55.4	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{-1} + x, 1\right) \]
      +-commutative [<=] 55.4	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{x + -1}, 1\right) \]

   3. Taylor expanded in y around inf 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ \left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right)\right) - \frac{1}{{y}^{2}} \]
      sub-neg [=>] 100.0	\[ \color{blue}{\left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right)\right) + \left(-\frac{1}{{y}^{2}}\right)} \]
      +-commutative [=>] 100.0	\[ \color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \frac{x}{{y}^{2}}\right)} + \left(-\frac{1}{{y}^{2}}\right) \]
      associate-+l+ [=>] 100.0	\[ \color{blue}{\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} + \left(-\frac{1}{{y}^{2}}\right)\right)} \]
      associate-+r+ [=>] 100.0	\[ \color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + x\right)} + \left(\frac{x}{{y}^{2}} + \left(-\frac{1}{{y}^{2}}\right)\right) \]
      associate-+l+ [=>] 100.0	\[ \color{blue}{\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \left(x + \left(\frac{x}{{y}^{2}} + \left(-\frac{1}{{y}^{2}}\right)\right)\right)} \]

   if -12000 < y < 6.1e9

   1. Initial program 99.8%

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

   if 6.1e9 < y

   1. Initial program 26.9%

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

   2. Simplified 54.5%

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

      [Start] 26.9	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 26.9	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 26.9	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      *-lft-identity [<=] 26.9	\[ \color{blue}{1 \cdot \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} + 1 \]
      associate-/l* [=>] 54.5	\[ 1 \cdot \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\right) + 1 \]
      distribute-neg-frac [=>] 54.5	\[ 1 \cdot \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*r/ [=>] 54.5	\[ \color{blue}{\frac{1 \cdot \left(-\left(1 - x\right)\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*l/ [<=] 54.5	\[ \color{blue}{\frac{1}{\frac{y + 1}{y}} \cdot \left(-\left(1 - x\right)\right)} + 1 \]
      fma-def [=>] 54.5	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{y + 1}{y}}, -\left(1 - x\right), 1\right)} \]
      associate-/l* [<=] 54.5	\[ \mathsf{fma}\left(\color{blue}{\frac{1 \cdot y}{y + 1}}, -\left(1 - x\right), 1\right) \]
      *-lft-identity [=>] 54.5	\[ \mathsf{fma}\left(\frac{\color{blue}{y}}{y + 1}, -\left(1 - x\right), 1\right) \]
      +-commutative [=>] 54.5	\[ \mathsf{fma}\left(\frac{y}{\color{blue}{1 + y}}, -\left(1 - x\right), 1\right) \]
      neg-sub0 [=>] 54.5	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{0 - \left(1 - x\right)}, 1\right) \]
      associate--r- [=>] 54.5	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{\left(0 - 1\right) + x}, 1\right) \]
      metadata-eval [=>] 54.5	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{-1} + x, 1\right) \]
      +-commutative [<=] 54.5	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{x + -1}, 1\right) \]

   3. Taylor expanded in y around inf 99.9%

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

   4. Simplified 99.9%

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

      [Start] 99.9	\[ -1 \cdot \frac{x - 1}{y} + x \]
      +-commutative [=>] 99.9	\[ \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
      mul-1-neg [=>] 99.9	\[ x + \color{blue}{\left(-\frac{x - 1}{y}\right)} \]
      unsub-neg [=>] 99.9	\[ \color{blue}{x - \frac{x - 1}{y}} \]
      sub-neg [=>] 99.9	\[ x - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 99.9	\[ x - \frac{x + \color{blue}{-1}}{y} \]

   5. Taylor expanded in x around 0 99.7%

      \[\leadsto x - \color{blue}{\frac{-1}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12000:\\ \;\;\;\;\frac{x + -1}{y} \cdot \left(-1 + \frac{-1}{y \cdot y}\right) + \left(x + \left(\frac{-1}{y \cdot y} + \frac{x}{y \cdot y}\right)\right)\\ \mathbf{elif}\;y \leq 6100000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;y \leq -200000:\\ \;\;\;\;\left(x + \frac{-1}{y \cdot y}\right) - \frac{x + -1}{y} \cdot \left(1 + \frac{1}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 185000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;y \leq -230000:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 30000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -85000000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{-1}{y}}{y}\\ \mathbf{elif}\;y \leq 11000000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4
Accuracy	99.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -260000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{-1}{y}}{y}\\ \mathbf{elif}\;y \leq 2200000000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 5
Accuracy	98.4%
Cost	836

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

Alternative 6
Accuracy	74.3%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-71}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-23}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	98.2%
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 8
Accuracy	98.3%
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 9
Accuracy	74.2%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.43:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	97.9%
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 11
Accuracy	86.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 320:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	74.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.95:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Accuracy	38.1%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023135 
(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))))