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

Average Error: 21.9 → 0.1

Time: 54.1s

Precision: binary64

Cost: 1096

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{y} + x\\ \mathbf{if}\;y \leq -2.25 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 85000000000000:\\ \;\;\;\;\frac{\left(-1 - y\right) - \left(-1 + x\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 y) x)))
   (if (<= y -2.25e+20)
     t_0
     (if (<= y 85000000000000.0)
       (/ (- (- -1.0 y) (* (+ -1.0 x) y)) (- -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 = (1.0 / y) + x;
	double tmp;
	if (y <= -2.25e+20) {
		tmp = t_0;
	} else if (y <= 85000000000000.0) {
		tmp = ((-1.0 - y) - ((-1.0 + x) * y)) / (-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 = (1.0d0 / y) + x
    if (y <= (-2.25d+20)) then
        tmp = t_0
    else if (y <= 85000000000000.0d0) then
        tmp = (((-1.0d0) - y) - (((-1.0d0) + x) * y)) / ((-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 = (1.0 / y) + x;
	double tmp;
	if (y <= -2.25e+20) {
		tmp = t_0;
	} else if (y <= 85000000000000.0) {
		tmp = ((-1.0 - y) - ((-1.0 + x) * y)) / (-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 = (1.0 / y) + x
	tmp = 0
	if y <= -2.25e+20:
		tmp = t_0
	elif y <= 85000000000000.0:
		tmp = ((-1.0 - y) - ((-1.0 + x) * y)) / (-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(1.0 / y) + x)
	tmp = 0.0
	if (y <= -2.25e+20)
		tmp = t_0;
	elseif (y <= 85000000000000.0)
		tmp = Float64(Float64(Float64(-1.0 - y) - Float64(Float64(-1.0 + x) * y)) / 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 = (1.0 / y) + x;
	tmp = 0.0;
	if (y <= -2.25e+20)
		tmp = t_0;
	elseif (y <= 85000000000000.0)
		tmp = ((-1.0 - y) - ((-1.0 + x) * y)) / (-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[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -2.25e+20], t$95$0, If[LessEqual[y, 85000000000000.0], N[(N[(N[(-1.0 - y), $MachinePrecision] - N[(N[(-1.0 + x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	21.9
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 < -2.25e20 or 8.5e13 < y

   1. Initial program 46.1

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

   2. Taylor expanded in y around -inf 0.0

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

   3. Simplified 0.0

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

   4. Taylor expanded in x around 0 0.0

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

   if -2.25e20 < y < 8.5e13

   1. Initial program 0.6

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

   2. Applied egg-rr 0.3

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

Alternatives

Alternative 1
Error	0.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -24000000:\\ \;\;\;\;\frac{1 - x}{y} + x\\ \mathbf{elif}\;y \leq 31500000000:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]

Alternative 2
Error	1.2
Cost	712

\[\begin{array}{l} t_0 := \frac{1 - x}{y} + x\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;1 - \left(-y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.1
Cost	712

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

Alternative 4
Error	1.4
Cost	648

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

Alternative 5
Error	17.1
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+84}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -0.000185:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 0.7:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	8.9
Cost	584

\[\begin{array}{l} t_0 := \frac{1}{y} + x\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.236:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	8.7
Cost	584

\[\begin{array}{l} t_0 := \frac{1}{y} + x\\ \mathbf{if}\;y \leq -6200000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3500000000000:\\ \;\;\;\;\frac{-1}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	16.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.58:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	16.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	38.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023033 
(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))))