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

Average Error: 22.0 → 0.1

Time: 9.7s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5200000 \lor \neg \left(y \leq 2350000\right):\\ \;\;\;\;x + \frac{\frac{-1}{y} + \left(1 - x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(-1 + x\right)}{y + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5200000.0) (not (<= y 2350000.0)))
   (+ x (/ (+ (/ -1.0 y) (- 1.0 x)) y))
   (+ 1.0 (/ (* y (+ -1.0 x)) (+ 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 <= -5200000.0) || !(y <= 2350000.0)) {
		tmp = x + (((-1.0 / y) + (1.0 - x)) / y);
	} else {
		tmp = 1.0 + ((y * (-1.0 + x)) / (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 <= (-5200000.0d0)) .or. (.not. (y <= 2350000.0d0))) then
        tmp = x + ((((-1.0d0) / y) + (1.0d0 - x)) / y)
    else
        tmp = 1.0d0 + ((y * ((-1.0d0) + x)) / (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 <= -5200000.0) || !(y <= 2350000.0)) {
		tmp = x + (((-1.0 / y) + (1.0 - x)) / y);
	} else {
		tmp = 1.0 + ((y * (-1.0 + x)) / (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 <= -5200000.0) or not (y <= 2350000.0):
		tmp = x + (((-1.0 / y) + (1.0 - x)) / y)
	else:
		tmp = 1.0 + ((y * (-1.0 + x)) / (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 <= -5200000.0) || !(y <= 2350000.0))
		tmp = Float64(x + Float64(Float64(Float64(-1.0 / y) + Float64(1.0 - x)) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(-1.0 + x)) / 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 <= -5200000.0) || ~((y <= 2350000.0)))
		tmp = x + (((-1.0 / y) + (1.0 - x)) / y);
	else
		tmp = 1.0 + ((y * (-1.0 + x)) / (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, -5200000.0], N[Not[LessEqual[y, 2350000.0]], $MachinePrecision]], N[(x + N[(N[(N[(-1.0 / y), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	22.0
Target	0.2
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -5.2e6 or 2.35e6 < y

   1. Initial program 44.9

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

   2. Simplified 28.6

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

      [Start] 44.9	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 44.9	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 44.9	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      neg-mul-1 [=>] 44.9	\[ \color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}} + 1 \]
      associate-*l/ [<=] 28.6	\[ -1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)} + 1 \]
      associate-*r* [=>] 28.6	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} + 1 \]
      fma-def [=>] 28.6	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{1 - x}{y + 1}, y, 1\right)} \]
      associate-*r/ [=>] 28.6	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{y + 1}}, y, 1\right) \]
      neg-mul-1 [<=] 28.6	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(1 - x\right)}}{y + 1}, y, 1\right) \]
      neg-sub0 [=>] 28.6	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 28.6	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 28.6	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [<=] 28.6	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 28.6	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Taylor expanded in y around -inf 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{y} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{y} \]
      associate--l+ [=>] 0.0	\[ \color{blue}{\frac{1}{y} + \left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right) + \frac{1}{y}} \]
      associate-+l- [=>] 0.0	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \left(\frac{x}{y} - \frac{1}{y}\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(x + -1 \cdot \frac{1 - x}{{y}^{2}}\right)} - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      associate-*r/ [=>] 0.0	\[ \left(x + \color{blue}{\frac{-1 \cdot \left(1 - x\right)}{{y}^{2}}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      sub-neg [=>] 0.0	\[ \left(x + \frac{-1 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-in [=>] 0.0	\[ \left(x + \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \left(-x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-commutative [<=] 0.0	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x\right) \cdot -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-neg-in [<=] 0.0	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x \cdot -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-neg-in [=>] 0.0	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{x \cdot \left(--1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      metadata-eval [=>] 0.0	\[ \left(x + \frac{-1 \cdot 1 + x \cdot \color{blue}{1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-in [<=] 0.0	\[ \left(x + \frac{\color{blue}{1 \cdot \left(-1 + x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [<=] 0.0	\[ \left(x + \frac{1 \cdot \color{blue}{\left(x + -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-lft-identity [=>] 0.0	\[ \left(x + \frac{\color{blue}{x + -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [=>] 0.0	\[ \left(x + \frac{\color{blue}{-1 + x}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      unpow2 [=>] 0.0	\[ \left(x + \frac{-1 + x}{\color{blue}{y \cdot y}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      div-sub [<=] 0.0	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \color{blue}{\frac{x - 1}{y}} \]
      sub-neg [=>] 0.0	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 0.0	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{x + \color{blue}{-1}}{y} \]
      +-commutative [=>] 0.0	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{-1 + x}}{y} \]

   5. Taylor expanded in x around 0 0.0

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

   6. Simplified 0.0

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

      [Start] 0.0	\[ \left(x + \frac{-1}{{y}^{2}}\right) - \frac{-1 + x}{y} \]
      unpow2 [=>] 0.0	\[ \left(x + \frac{-1}{\color{blue}{y \cdot y}}\right) - \frac{-1 + x}{y} \]

   7. Applied egg-rr 0.0

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

   if -5.2e6 < y < 2.35e6

   1. Initial program 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5200000 \lor \neg \left(y \leq 2350000\right):\\ \;\;\;\;x + \frac{\frac{-1}{y} + \left(1 - x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(-1 + x\right)}{y + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	969

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

Alternative 2
Error	0.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -3900000 \lor \neg \left(y \leq 2800000\right):\\ \;\;\;\;x + \frac{\frac{-1}{y} + \left(1 - x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-1 + x}{y + 1}\\ \end{array} \]

Alternative 3
Error	8.9
Cost	716

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-41}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 0.01:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5
Error	1.0
Cost	713

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

Alternative 6
Error	1.0
Cost	712

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

Alternative 7
Error	16.2
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	16.2
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-39}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	1.3
Cost	585

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

Alternative 10
Error	16.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	39.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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))))