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

Average Error: 22.6 → 0.0

Time: 17.7s

Precision: binary64

Cost: 7816

Math

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

↓

\[\begin{array}{l} t_0 := \left(x + \frac{1}{y}\right) + \left(\left(-\frac{1 + \left(-x\right)}{{y}^{2}}\right) - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -260000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 340000:\\ \;\;\;\;1 - y \cdot \frac{1 - x}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+ (+ x (/ 1.0 y)) (- (- (/ (+ 1.0 (- x)) (pow y 2.0))) (/ x y)))))
   (if (<= y -260000.0)
     t_0
     (if (<= y 340000.0) (- 1.0 (* y (/ (- 1.0 x) (+ 1.0 y)))) t_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 = (x + (1.0 / y)) + (-((1.0 + -x) / pow(y, 2.0)) - (x / y));
	double tmp;
	if (y <= -260000.0) {
		tmp = t_0;
	} else if (y <= 340000.0) {
		tmp = 1.0 - (y * ((1.0 - x) / (1.0 + y)));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (x + (1.0d0 / y)) + (-((1.0d0 + -x) / (y ** 2.0d0)) - (x / y))
    if (y <= (-260000.0d0)) then
        tmp = t_0
    else if (y <= 340000.0d0) then
        tmp = 1.0d0 - (y * ((1.0d0 - x) / (1.0d0 + y)))
    else
        tmp = t_0
    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 = (x + (1.0 / y)) + (-((1.0 + -x) / Math.pow(y, 2.0)) - (x / y));
	double tmp;
	if (y <= -260000.0) {
		tmp = t_0;
	} else if (y <= 340000.0) {
		tmp = 1.0 - (y * ((1.0 - x) / (1.0 + y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = (x + (1.0 / y)) + (-((1.0 + -x) / math.pow(y, 2.0)) - (x / y))
	tmp = 0
	if y <= -260000.0:
		tmp = t_0
	elif y <= 340000.0:
		tmp = 1.0 - (y * ((1.0 - x) / (1.0 + y)))
	else:
		tmp = t_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(x + Float64(1.0 / y)) + Float64(Float64(-Float64(Float64(1.0 + Float64(-x)) / (y ^ 2.0))) - Float64(x / y)))
	tmp = 0.0
	if (y <= -260000.0)
		tmp = t_0;
	elseif (y <= 340000.0)
		tmp = Float64(1.0 - Float64(y * Float64(Float64(1.0 - x) / Float64(1.0 + y))));
	else
		tmp = t_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)
	t_0 = (x + (1.0 / y)) + (-((1.0 + -x) / (y ^ 2.0)) - (x / y));
	tmp = 0.0;
	if (y <= -260000.0)
		tmp = t_0;
	elseif (y <= 340000.0)
		tmp = 1.0 - (y * ((1.0 - x) / (1.0 + y)));
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + N[((-N[(N[(1.0 + (-x)), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]) - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -260000.0], t$95$0, If[LessEqual[y, 340000.0], N[(1.0 - N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	22.6
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 < -2.6e5 or 3.4e5 < y

   1. Initial program 45.8

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

   2. Simplified 29.6

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

      [Start] 45.8	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      rational.json-simplify-49 [=>] 29.6	\[ 1 - \color{blue}{y \cdot \frac{1 - x}{y + 1}} \]
      rational.json-simplify-1 [=>] 29.6	\[ 1 - y \cdot \frac{1 - x}{\color{blue}{1 + y}} \]
      rational.json-simplify-17 [=>] 29.6	\[ 1 - y \cdot \frac{1 - x}{\color{blue}{y - -1}} \]
      rational.json-simplify-50 [=>] 29.6	\[ 1 - y \cdot \color{blue}{\frac{-\left(1 - x\right)}{-1 - y}} \]
      rational.json-simplify-8 [=>] 29.6	\[ 1 - y \cdot \frac{\color{blue}{\left(1 - x\right) \cdot -1}}{-1 - y} \]
      rational.json-simplify-2 [=>] 29.6	\[ 1 - y \cdot \frac{\color{blue}{-1 \cdot \left(1 - x\right)}}{-1 - y} \]
      rational.json-simplify-49 [=>] 29.7	\[ 1 - y \cdot \color{blue}{\left(\left(1 - x\right) \cdot \frac{-1}{-1 - y}\right)} \]
      rational.json-simplify-2 [=>] 29.7	\[ 1 - y \cdot \color{blue}{\left(\frac{-1}{-1 - y} \cdot \left(1 - x\right)\right)} \]
      rational.json-simplify-2 [<=] 29.7	\[ 1 - y \cdot \color{blue}{\left(\left(1 - x\right) \cdot \frac{-1}{-1 - y}\right)} \]
      rational.json-simplify-49 [<=] 29.6	\[ 1 - y \cdot \color{blue}{\frac{-1 \cdot \left(1 - x\right)}{-1 - y}} \]
      rational.json-simplify-2 [<=] 29.6	\[ 1 - y \cdot \frac{\color{blue}{\left(1 - x\right) \cdot -1}}{-1 - y} \]
      rational.json-simplify-8 [<=] 29.6	\[ 1 - y \cdot \frac{\color{blue}{-\left(1 - x\right)}}{-1 - y} \]
      rational.json-simplify-50 [<=] 29.6	\[ 1 - y \cdot \color{blue}{\frac{1 - x}{y - -1}} \]
      rational.json-simplify-17 [<=] 29.6	\[ 1 - y \cdot \frac{1 - x}{\color{blue}{1 + y}} \]

   3. Taylor expanded in y around inf 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{y} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right)\right) - \frac{x}{y} \]
      rational.json-simplify-41 [=>] 0.0	\[ \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + \left(x + \frac{1}{y}\right)\right)} - \frac{x}{y} \]
      rational.json-simplify-1 [<=] 0.0	\[ \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + \color{blue}{\left(\frac{1}{y} + x\right)}\right) - \frac{x}{y} \]
      rational.json-simplify-48 [=>] 0.0	\[ \color{blue}{\left(\frac{1}{y} + x\right) + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} - \frac{x}{y}\right)} \]
      rational.json-simplify-1 [=>] 0.0	\[ \color{blue}{\left(x + \frac{1}{y}\right)} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} - \frac{x}{y}\right) \]
      rational.json-simplify-2 [=>] 0.0	\[ \left(x + \frac{1}{y}\right) + \left(\color{blue}{\frac{1 + -1 \cdot x}{{y}^{2}} \cdot -1} - \frac{x}{y}\right) \]
      rational.json-simplify-9 [=>] 0.0	\[ \left(x + \frac{1}{y}\right) + \left(\color{blue}{\left(-\frac{1 + -1 \cdot x}{{y}^{2}}\right)} - \frac{x}{y}\right) \]
      rational.json-simplify-2 [=>] 0.0	\[ \left(x + \frac{1}{y}\right) + \left(\left(-\frac{1 + \color{blue}{x \cdot -1}}{{y}^{2}}\right) - \frac{x}{y}\right) \]
      rational.json-simplify-9 [=>] 0.0	\[ \left(x + \frac{1}{y}\right) + \left(\left(-\frac{1 + \color{blue}{\left(-x\right)}}{{y}^{2}}\right) - \frac{x}{y}\right) \]

   if -2.6e5 < y < 3.4e5

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      rational.json-simplify-49 [=>] 0.1	\[ 1 - \color{blue}{y \cdot \frac{1 - x}{y + 1}} \]
      rational.json-simplify-1 [=>] 0.1	\[ 1 - y \cdot \frac{1 - x}{\color{blue}{1 + y}} \]
      rational.json-simplify-17 [=>] 0.1	\[ 1 - y \cdot \frac{1 - x}{\color{blue}{y - -1}} \]
      rational.json-simplify-50 [=>] 0.1	\[ 1 - y \cdot \color{blue}{\frac{-\left(1 - x\right)}{-1 - y}} \]
      rational.json-simplify-8 [=>] 0.1	\[ 1 - y \cdot \frac{\color{blue}{\left(1 - x\right) \cdot -1}}{-1 - y} \]
      rational.json-simplify-2 [=>] 0.1	\[ 1 - y \cdot \frac{\color{blue}{-1 \cdot \left(1 - x\right)}}{-1 - y} \]
      rational.json-simplify-49 [=>] 0.1	\[ 1 - y \cdot \color{blue}{\left(\left(1 - x\right) \cdot \frac{-1}{-1 - y}\right)} \]
      rational.json-simplify-2 [=>] 0.1	\[ 1 - y \cdot \color{blue}{\left(\frac{-1}{-1 - y} \cdot \left(1 - x\right)\right)} \]
      rational.json-simplify-2 [<=] 0.1	\[ 1 - y \cdot \color{blue}{\left(\left(1 - x\right) \cdot \frac{-1}{-1 - y}\right)} \]
      rational.json-simplify-49 [<=] 0.1	\[ 1 - y \cdot \color{blue}{\frac{-1 \cdot \left(1 - x\right)}{-1 - y}} \]
      rational.json-simplify-2 [<=] 0.1	\[ 1 - y \cdot \frac{\color{blue}{\left(1 - x\right) \cdot -1}}{-1 - y} \]
      rational.json-simplify-8 [<=] 0.1	\[ 1 - y \cdot \frac{\color{blue}{-\left(1 - x\right)}}{-1 - y} \]
      rational.json-simplify-50 [<=] 0.1	\[ 1 - y \cdot \color{blue}{\frac{1 - x}{y - -1}} \]
      rational.json-simplify-17 [<=] 0.1	\[ 1 - y \cdot \frac{1 - x}{\color{blue}{1 + y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260000:\\ \;\;\;\;\left(x + \frac{1}{y}\right) + \left(\left(-\frac{1 + \left(-x\right)}{{y}^{2}}\right) - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 340000:\\ \;\;\;\;1 - y \cdot \frac{1 - x}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) + \left(\left(-\frac{1 + \left(-x\right)}{{y}^{2}}\right) - \frac{x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	968

\[\begin{array}{l} t_0 := x + \left(-\frac{x + -1}{y}\right)\\ \mathbf{if}\;y \leq -175000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 130000000:\\ \;\;\;\;1 - y \cdot \frac{1 - x}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	9.2
Cost	848

\[\begin{array}{l} t_0 := \frac{1}{y} + x\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-100}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-79}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.1
Cost	848

\[\begin{array}{l} t_0 := \frac{1}{y} + x\\ \mathbf{if}\;y \leq -21000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{1 + y}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-80}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.0
Cost	776

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

Alternative 5
Error	16.7
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-79}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	1.2
Cost	712

\[\begin{array}{l} t_0 := \frac{1}{y} + x\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.78:\\ \;\;\;\;1 - \left(1 - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.4
Cost	648

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

Alternative 8
Error	16.8
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-101}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-79}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	16.6
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	39.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023069 
(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))))