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

Average Accuracy: 64.9% → 99.4%

Time: 17.9s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+22}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 200000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -2.45e+22)
   (- x (/ -1.0 y))
   (if (<= y 200000000.0)
     (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))
     (+ x (/ (- 1.0 x) 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 tmp;
	if (y <= -2.45e+22) {
		tmp = x - (-1.0 / y);
	} else if (y <= 200000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x + ((1.0 - x) / 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) :: tmp
    if (y <= (-2.45d+22)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 200000000.0d0) then
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    else
        tmp = x + ((1.0d0 - x) / 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 tmp;
	if (y <= -2.45e+22) {
		tmp = x - (-1.0 / y);
	} else if (y <= 200000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	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 <= -2.45e+22:
		tmp = x - (-1.0 / y)
	elif y <= 200000000.0:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	else:
		tmp = x + ((1.0 - x) / 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)
	tmp = 0.0
	if (y <= -2.45e+22)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 200000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / 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)
	tmp = 0.0;
	if (y <= -2.45e+22)
		tmp = x - (-1.0 / y);
	elseif (y <= 200000000.0)
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	else
		tmp = x + ((1.0 - x) / 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_] := If[LessEqual[y, -2.45e+22], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 200000000.0], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	64.9%
Target	99.6%
Herbie	99.4%

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

Derivation

1. Split input into 3 regimes

2. if y < -2.4499999999999999e22

   1. Initial program 26.2%

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

   2. Simplified 52.9%

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

      [Start] 26.2	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 26.2	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      distribute-neg-frac [=>] 26.2	\[ 1 + \color{blue}{\frac{-\left(1 - x\right) \cdot y}{y + 1}} \]
      neg-mul-1 [=>] 26.2	\[ 1 + \frac{\color{blue}{-1 \cdot \left(\left(1 - x\right) \cdot y\right)}}{y + 1} \]
      associate-*l/ [<=] 26.0	\[ 1 + \color{blue}{\frac{-1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)} \]
      metadata-eval [<=] 26.0	\[ 1 + \frac{\color{blue}{1 \cdot -1}}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      associate-*l/ [<=] 26.0	\[ 1 + \color{blue}{\left(\frac{1}{y + 1} \cdot -1\right)} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      associate-/r/ [<=] 26.0	\[ 1 + \color{blue}{\frac{1}{\frac{y + 1}{-1}}} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      metadata-eval [<=] 26.0	\[ 1 + \frac{\color{blue}{--1}}{\frac{y + 1}{-1}} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      distribute-neg-frac [<=] 26.0	\[ 1 + \color{blue}{\left(-\frac{-1}{\frac{y + 1}{-1}}\right)} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      cancel-sign-sub-inv [<=] 26.0	\[ \color{blue}{1 - \frac{-1}{\frac{y + 1}{-1}} \cdot \left(\left(1 - x\right) \cdot y\right)} \]
      associate-/r/ [<=] 26.1	\[ 1 - \color{blue}{\frac{-1}{\frac{\frac{y + 1}{-1}}{\left(1 - x\right) \cdot y}}} \]
      associate-/r* [<=] 26.1	\[ 1 - \frac{-1}{\color{blue}{\frac{y + 1}{-1 \cdot \left(\left(1 - x\right) \cdot y\right)}}} \]
      neg-mul-1 [<=] 26.1	\[ 1 - \frac{-1}{\frac{y + 1}{\color{blue}{-\left(1 - x\right) \cdot y}}} \]
      associate-/r/ [=>] 26.0	\[ 1 - \color{blue}{\frac{-1}{y + 1} \cdot \left(-\left(1 - x\right) \cdot y\right)} \]
      distribute-rgt-neg-in [<=] 26.0	\[ 1 - \color{blue}{\left(-\frac{-1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\right)} \]
      associate-/r/ [<=] 26.1	\[ 1 - \left(-\color{blue}{\frac{-1}{\frac{y + 1}{\left(1 - x\right) \cdot y}}}\right) \]
      distribute-neg-frac [=>] 26.1	\[ 1 - \color{blue}{\frac{--1}{\frac{y + 1}{\left(1 - x\right) \cdot y}}} \]
      metadata-eval [=>] 26.1	\[ 1 - \frac{\color{blue}{1}}{\frac{y + 1}{\left(1 - x\right) \cdot y}} \]
      associate-/r/ [=>] 26.0	\[ 1 - \color{blue}{\frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)} \]

   3. Taylor expanded in y around inf 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ \left(\frac{1}{y} + x\right) - \frac{x}{y} \]
      +-commutative [=>] 100.0	\[ \color{blue}{\left(x + \frac{1}{y}\right)} - \frac{x}{y} \]
      associate--l+ [=>] 100.0	\[ \color{blue}{x + \left(\frac{1}{y} - \frac{x}{y}\right)} \]
      div-sub [<=] 100.0	\[ x + \color{blue}{\frac{1 - x}{y}} \]
      sub-neg [=>] 100.0	\[ x + \frac{\color{blue}{1 + \left(-x\right)}}{y} \]
      +-commutative [=>] 100.0	\[ x + \frac{\color{blue}{\left(-x\right) + 1}}{y} \]
      neg-sub0 [=>] 100.0	\[ x + \frac{\color{blue}{\left(0 - x\right)} + 1}{y} \]
      associate-+l- [=>] 100.0	\[ x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]
      neg-sub0 [<=] 100.0	\[ x + \frac{\color{blue}{-\left(x - 1\right)}}{y} \]
      mul-1-neg [<=] 100.0	\[ x + \frac{\color{blue}{-1 \cdot \left(x - 1\right)}}{y} \]
      associate-*r/ [<=] 100.0	\[ x + \color{blue}{-1 \cdot \frac{x - 1}{y}} \]
      mul-1-neg [=>] 100.0	\[ x + \color{blue}{\left(-\frac{x - 1}{y}\right)} \]
      unsub-neg [=>] 100.0	\[ \color{blue}{x - \frac{x - 1}{y}} \]
      sub-neg [=>] 100.0	\[ x - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 100.0	\[ x - \frac{x + \color{blue}{-1}}{y} \]

   5. Taylor expanded in x around 0 100.0%

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

   if -2.4499999999999999e22 < y < 2e8

   1. Initial program 99.0%

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

   if 2e8 < y

   1. Initial program 30.0%

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

   2. Simplified 54.1%

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

      [Start] 30.0	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 30.0	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      distribute-neg-frac [=>] 30.0	\[ 1 + \color{blue}{\frac{-\left(1 - x\right) \cdot y}{y + 1}} \]
      neg-mul-1 [=>] 30.0	\[ 1 + \frac{\color{blue}{-1 \cdot \left(\left(1 - x\right) \cdot y\right)}}{y + 1} \]
      associate-*l/ [<=] 30.0	\[ 1 + \color{blue}{\frac{-1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)} \]
      metadata-eval [<=] 30.0	\[ 1 + \frac{\color{blue}{1 \cdot -1}}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      associate-*l/ [<=] 30.0	\[ 1 + \color{blue}{\left(\frac{1}{y + 1} \cdot -1\right)} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      associate-/r/ [<=] 30.0	\[ 1 + \color{blue}{\frac{1}{\frac{y + 1}{-1}}} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      metadata-eval [<=] 30.0	\[ 1 + \frac{\color{blue}{--1}}{\frac{y + 1}{-1}} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      distribute-neg-frac [<=] 30.0	\[ 1 + \color{blue}{\left(-\frac{-1}{\frac{y + 1}{-1}}\right)} \cdot \left(\left(1 - x\right) \cdot y\right) \]
      cancel-sign-sub-inv [<=] 30.0	\[ \color{blue}{1 - \frac{-1}{\frac{y + 1}{-1}} \cdot \left(\left(1 - x\right) \cdot y\right)} \]
      associate-/r/ [<=] 29.9	\[ 1 - \color{blue}{\frac{-1}{\frac{\frac{y + 1}{-1}}{\left(1 - x\right) \cdot y}}} \]
      associate-/r* [<=] 29.9	\[ 1 - \frac{-1}{\color{blue}{\frac{y + 1}{-1 \cdot \left(\left(1 - x\right) \cdot y\right)}}} \]
      neg-mul-1 [<=] 29.9	\[ 1 - \frac{-1}{\frac{y + 1}{\color{blue}{-\left(1 - x\right) \cdot y}}} \]
      associate-/r/ [=>] 30.0	\[ 1 - \color{blue}{\frac{-1}{y + 1} \cdot \left(-\left(1 - x\right) \cdot y\right)} \]
      distribute-rgt-neg-in [<=] 30.0	\[ 1 - \color{blue}{\left(-\frac{-1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\right)} \]
      associate-/r/ [<=] 29.9	\[ 1 - \left(-\color{blue}{\frac{-1}{\frac{y + 1}{\left(1 - x\right) \cdot y}}}\right) \]
      distribute-neg-frac [=>] 29.9	\[ 1 - \color{blue}{\frac{--1}{\frac{y + 1}{\left(1 - x\right) \cdot y}}} \]
      metadata-eval [=>] 29.9	\[ 1 - \frac{\color{blue}{1}}{\frac{y + 1}{\left(1 - x\right) \cdot y}} \]
      associate-/r/ [=>] 30.0	\[ 1 - \color{blue}{\frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)} \]

   3. Taylor expanded in y around inf 99.7%

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

   4. Simplified 99.7%

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

      [Start] 99.7	\[ \left(\frac{1}{y} + x\right) - \frac{x}{y} \]
      +-commutative [=>] 99.7	\[ \color{blue}{\left(x + \frac{1}{y}\right)} - \frac{x}{y} \]
      associate--l+ [=>] 99.7	\[ \color{blue}{x + \left(\frac{1}{y} - \frac{x}{y}\right)} \]
      div-sub [<=] 99.7	\[ x + \color{blue}{\frac{1 - x}{y}} \]
      sub-neg [=>] 99.7	\[ x + \frac{\color{blue}{1 + \left(-x\right)}}{y} \]
      +-commutative [=>] 99.7	\[ x + \frac{\color{blue}{\left(-x\right) + 1}}{y} \]
      neg-sub0 [=>] 99.7	\[ x + \frac{\color{blue}{\left(0 - x\right)} + 1}{y} \]
      associate-+l- [=>] 99.7	\[ x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]
      neg-sub0 [<=] 99.7	\[ x + \frac{\color{blue}{-\left(x - 1\right)}}{y} \]
      mul-1-neg [<=] 99.7	\[ x + \frac{\color{blue}{-1 \cdot \left(x - 1\right)}}{y} \]
      associate-*r/ [<=] 99.7	\[ x + \color{blue}{-1 \cdot \frac{x - 1}{y}} \]
      mul-1-neg [=>] 99.7	\[ x + \color{blue}{\left(-\frac{x - 1}{y}\right)} \]
      unsub-neg [=>] 99.7	\[ \color{blue}{x - \frac{x - 1}{y}} \]
      sub-neg [=>] 99.7	\[ x - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 99.7	\[ x - \frac{x + \color{blue}{-1}}{y} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+22}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 200000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+22}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 200000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 2
Accuracy	98.1%
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 3
Accuracy	98.2%
Cost	712

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

Alternative 4
Accuracy	85.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.05 \cdot 10^{-7}\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 5
Accuracy	97.8%
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	73.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	73.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1300:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	38.3%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023146 
(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))))