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

Percentage Accurate: 87.7% → 99.4%

Time: 9.2s

Precision: binary64

Cost: 841

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+94} \lor \neg \left(z \leq 1.4 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.1e+94) (not (<= z 1.4e+16)))
   (- (* (/ y z) x) 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 <= -2.1e+94) || !(z <= 1.4e+16)) {
		tmp = ((y / z) * x) - 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 <= (-2.1d+94)) .or. (.not. (z <= 1.4d+16))) then
        tmp = ((y / z) * x) - 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 <= -2.1e+94) || !(z <= 1.4e+16)) {
		tmp = ((y / z) * x) - 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 <= -2.1e+94) or not (z <= 1.4e+16):
		tmp = ((y / z) * x) - 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 <= -2.1e+94) || !(z <= 1.4e+16))
		tmp = Float64(Float64(Float64(y / z) * x) - 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 <= -2.1e+94) || ~((z <= 1.4e+16)))
		tmp = ((y / z) * x) - 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, -2.1e+94], N[Not[LessEqual[z, 1.4e+16]], $MachinePrecision]], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Target

Original	87.7%
Target	99.4%
Herbie	99.4%

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

Derivation

1. Split input into 2 regimes

2. if z < -2.09999999999999989e94 or 1.4e16 < z

   1. Initial program 76.4%

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

   2. Simplified 93.6%

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

      [Start] 76.4%	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      +-commutative [=>] 76.4%	\[ \frac{x \cdot \color{blue}{\left(1 + \left(y - z\right)\right)}}{z} \]
      sub-neg [=>] 76.4%	\[ \frac{x \cdot \left(1 + \color{blue}{\left(y + \left(-z\right)\right)}\right)}{z} \]
      +-commutative [=>] 76.4%	\[ \frac{x \cdot \left(1 + \color{blue}{\left(\left(-z\right) + y\right)}\right)}{z} \]
      associate-+r+ [=>] 76.4%	\[ \frac{x \cdot \color{blue}{\left(\left(1 + \left(-z\right)\right) + y\right)}}{z} \]
      unsub-neg [=>] 76.4%	\[ \frac{x \cdot \left(\color{blue}{\left(1 - z\right)} + y\right)}{z} \]
      associate-+l- [=>] 76.4%	\[ \frac{x \cdot \color{blue}{\left(1 - \left(z - y\right)\right)}}{z} \]
      distribute-lft-out-- [<=] 76.4%	\[ \frac{\color{blue}{x \cdot 1 - x \cdot \left(z - y\right)}}{z} \]
      *-rgt-identity [=>] 76.4%	\[ \frac{\color{blue}{x} - x \cdot \left(z - y\right)}{z} \]
      distribute-rgt-out-- [<=] 76.1%	\[ \frac{x - \color{blue}{\left(z \cdot x - y \cdot x\right)}}{z} \]
      sub-neg [=>] 76.1%	\[ \frac{x - \color{blue}{\left(z \cdot x + \left(-y \cdot x\right)\right)}}{z} \]
      +-commutative [=>] 76.1%	\[ \frac{x - \color{blue}{\left(\left(-y \cdot x\right) + z \cdot x\right)}}{z} \]
      associate--r+ [=>] 76.1%	\[ \frac{\color{blue}{\left(x - \left(-y \cdot x\right)\right) - z \cdot x}}{z} \]
      div-sub [=>] 76.1%	\[ \color{blue}{\frac{x - \left(-y \cdot x\right)}{z} - \frac{z \cdot x}{z}} \]

   3. Taylor expanded in y around inf 93.6%

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

   4. Simplified 100.0%

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

      [Start] 93.6%	\[ \frac{y \cdot x}{z} - x \]
      associate-/l* [=>] 93.8%	\[ \color{blue}{\frac{y}{\frac{z}{x}}} - x \]
      associate-/r/ [=>] 100.0%	\[ \color{blue}{\frac{y}{z} \cdot x} - x \]

   if -2.09999999999999989e94 < z < 1.4e16

   1. Initial program 98.1%

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

   2. Simplified 91.4%

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

      [Start] 98.1%	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 91.4%	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   3. Taylor expanded in x around 0 98.1%

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

   4. Simplified 99.9%

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

      [Start] 98.1%	\[ \frac{x \cdot \left(\left(1 + y\right) - z\right)}{z} \]
      associate-*l/ [<=] 99.9%	\[ \color{blue}{\frac{x}{z} \cdot \left(\left(1 + y\right) - z\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+94} \lor \neg \left(z \leq 1.4 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+94} \lor \neg \left(z \leq 1.4 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \end{array} \]

Alternative 3
Accuracy	98.1%
Cost	1220

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq 10^{+21}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternative 4
Accuracy	65.9%
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+63}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-229}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 110000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 5
Accuracy	65.5%
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+63}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 14000000000:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Accuracy	98.8%
Cost	713

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

Alternative 7
Accuracy	86.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+63}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8
Accuracy	85.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+17} \lor \neg \left(y \leq 51600\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 9
Accuracy	85.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -35000000000000 \lor \neg \left(y \leq 51600\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 10
Accuracy	65.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -11.6:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2300000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 11
Accuracy	39.0%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023271 
(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))