Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.2% → 99.1%

Time: 5.7s

Precision: binary64

Cost: 841

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+107} \lor \neg \left(z \leq 7.5 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.15e+107) (not (<= z 7.5e+15)))
   (- (* x (/ y z)) x)
   (* (/ x z) (- (+ y 1.0) z))))

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e+107) || !(z <= 7.5e+15)) {
		tmp = (x * (y / z)) - x;
	} else {
		tmp = (x / z) * ((y + 1.0) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * ((y - z) + 1.0d0)) / z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.15d+107)) .or. (.not. (z <= 7.5d+15))) then
        tmp = (x * (y / z)) - x
    else
        tmp = (x / z) * ((y + 1.0d0) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e+107) || !(z <= 7.5e+15)) {
		tmp = (x * (y / z)) - x;
	} else {
		tmp = (x / z) * ((y + 1.0) - z);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (z <= -1.15e+107) or not (z <= 7.5e+15):
		tmp = (x * (y / z)) - x
	else:
		tmp = (x / z) * ((y + 1.0) - z)
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end

↓

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.15e+107) || !(z <= 7.5e+15))
		tmp = Float64(Float64(x * Float64(y / z)) - x);
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y + 1.0) - z));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * ((y - z) + 1.0)) / z;
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.15e+107) || ~((z <= 7.5e+15)))
		tmp = (x * (y / z)) - x;
	else
		tmp = (x / z) * ((y + 1.0) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := If[Or[LessEqual[z, -1.15e+107], N[Not[LessEqual[z, 7.5e+15]], $MachinePrecision]], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Target

Original	88.2%
Target	99.4%
Herbie	99.1%

\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e107 or 7.5e15 < z

   1. Initial program 72.9%

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

   2. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      Step-by-step derivation

      [Start] 72.9%	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      +-commutative [=>] 72.9%	\[ \frac{x \cdot \color{blue}{\left(1 + \left(y - z\right)\right)}}{z} \]
      sub-neg [=>] 72.9%	\[ \frac{x \cdot \left(1 + \color{blue}{\left(y + \left(-z\right)\right)}\right)}{z} \]
      +-commutative [=>] 72.9%	\[ \frac{x \cdot \left(1 + \color{blue}{\left(\left(-z\right) + y\right)}\right)}{z} \]
      associate-+r+ [=>] 72.9%	\[ \frac{x \cdot \color{blue}{\left(\left(1 + \left(-z\right)\right) + y\right)}}{z} \]
      unsub-neg [=>] 72.9%	\[ \frac{x \cdot \left(\color{blue}{\left(1 - z\right)} + y\right)}{z} \]
      associate-+l- [=>] 72.9%	\[ \frac{x \cdot \color{blue}{\left(1 - \left(z - y\right)\right)}}{z} \]
      distribute-lft-out-- [<=] 72.9%	\[ \frac{\color{blue}{x \cdot 1 - x \cdot \left(z - y\right)}}{z} \]
      *-rgt-identity [=>] 72.9%	\[ \frac{\color{blue}{x} - x \cdot \left(z - y\right)}{z} \]
      distribute-rgt-out-- [<=] 70.7%	\[ \frac{x - \color{blue}{\left(z \cdot x - y \cdot x\right)}}{z} \]
      sub-neg [=>] 70.7%	\[ \frac{x - \color{blue}{\left(z \cdot x + \left(-y \cdot x\right)\right)}}{z} \]
      +-commutative [=>] 70.7%	\[ \frac{x - \color{blue}{\left(\left(-y \cdot x\right) + z \cdot x\right)}}{z} \]
      associate--r+ [=>] 70.7%	\[ \frac{\color{blue}{\left(x - \left(-y \cdot x\right)\right) - z \cdot x}}{z} \]
      div-sub [=>] 70.7%	\[ \color{blue}{\frac{x - \left(-y \cdot x\right)}{z} - \frac{z \cdot x}{z}} \]

   3. Taylor expanded in y around inf 86.3%

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

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} - x \]
      Step-by-step derivation

      [Start] 86.3%	\[ \frac{y \cdot x}{z} - x \]
      associate-/l* [=>] 97.1%	\[ \color{blue}{\frac{y}{\frac{z}{x}}} - x \]
      associate-/r/ [=>] 99.9%	\[ \color{blue}{\frac{y}{z} \cdot x} - x \]

   if -1.15e107 < z < 7.5e15

   1. Initial program 99.2%

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

   2. Simplified 93.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]
      Step-by-step derivation

      [Start] 99.2%	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 93.4%	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   3. Taylor expanded in x around 0 99.2%

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

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(\left(1 + y\right) - z\right)} \]
      Step-by-step derivation

      [Start] 99.2%	\[ \frac{x \cdot \left(\left(1 + y\right) - z\right)}{z} \]
      associate-*l/ [<=] 99.8%	\[ \color{blue}{\frac{x}{z} \cdot \left(\left(1 + y\right) - z\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+107} \lor \neg \left(z \leq 7.5 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+107} \lor \neg \left(z \leq 7.5 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternative 2
Accuracy	98.1%
Cost	1220

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternative 3
Accuracy	64.3%
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Accuracy	94.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+20} \lor \neg \left(y \leq 2.7 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 5
Accuracy	96.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+20} \lor \neg \left(y \leq 2.7 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 6
Accuracy	98.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.98 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]

Alternative 7
Accuracy	85.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+54} \lor \neg \left(y \leq 1400000\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8
Accuracy	65.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0135:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9
Accuracy	38.7%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))