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

Average Accuracy: 83.3% → 99.8%

Time: 8.8s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \left(y - z\right)\\ \mathbf{if}\;z \leq -5800000000000:\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- y z))))
   (if (<= z -5800000000000.0)
     (* (+ (/ y z) -1.0) x)
     (if (<= z 1.6e+32) (/ (* x t_0) z) (/ x (/ z t_0))))))

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 t_0 = 1.0 + (y - z);
	double tmp;
	if (z <= -5800000000000.0) {
		tmp = ((y / z) + -1.0) * x;
	} else if (z <= 1.6e+32) {
		tmp = (x * t_0) / z;
	} else {
		tmp = x / (z / t_0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y - z)
    if (z <= (-5800000000000.0d0)) then
        tmp = ((y / z) + (-1.0d0)) * x
    else if (z <= 1.6d+32) then
        tmp = (x * t_0) / z
    else
        tmp = x / (z / t_0)
    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 t_0 = 1.0 + (y - z);
	double tmp;
	if (z <= -5800000000000.0) {
		tmp = ((y / z) + -1.0) * x;
	} else if (z <= 1.6e+32) {
		tmp = (x * t_0) / z;
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = 1.0 + (y - z)
	tmp = 0
	if z <= -5800000000000.0:
		tmp = ((y / z) + -1.0) * x
	elif z <= 1.6e+32:
		tmp = (x * t_0) / z
	else:
		tmp = x / (z / t_0)
	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)
	t_0 = Float64(1.0 + Float64(y - z))
	tmp = 0.0
	if (z <= -5800000000000.0)
		tmp = Float64(Float64(Float64(y / z) + -1.0) * x);
	elseif (z <= 1.6e+32)
		tmp = Float64(Float64(x * t_0) / z);
	else
		tmp = Float64(x / Float64(z / t_0));
	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)
	t_0 = 1.0 + (y - z);
	tmp = 0.0;
	if (z <= -5800000000000.0)
		tmp = ((y / z) + -1.0) * x;
	elseif (z <= 1.6e+32)
		tmp = (x * t_0) / z;
	else
		tmp = x / (z / t_0);
	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_] := Block[{t$95$0 = N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5800000000000.0], N[(N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.6e+32], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	83.3%
Target	99.3%
Herbie	99.8%

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

Derivation

1. Split input into 3 regimes

2. if z < -5.8e12

   1. Initial program 71.6%

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

   2. Simplified 90.3%

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

      [Start] 71.6	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-*r/ [<=] 99.9	\[ \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]
      +-commutative [=>] 99.9	\[ x \cdot \frac{\color{blue}{1 + \left(y - z\right)}}{z} \]
      associate-+r- [=>] 99.9	\[ x \cdot \frac{\color{blue}{\left(1 + y\right) - z}}{z} \]
      div-sub [=>] 99.9	\[ x \cdot \color{blue}{\left(\frac{1 + y}{z} - \frac{z}{z}\right)} \]
      *-inverses [=>] 99.9	\[ x \cdot \left(\frac{1 + y}{z} - \color{blue}{1}\right) \]
      distribute-rgt-out-- [<=] 99.9	\[ \color{blue}{\frac{1 + y}{z} \cdot x - 1 \cdot x} \]
      *-lft-identity [=>] 99.9	\[ \frac{1 + y}{z} \cdot x - \color{blue}{x} \]
      *-commutative [=>] 99.9	\[ \color{blue}{x \cdot \frac{1 + y}{z}} - x \]
      associate-*r/ [=>] 90.3	\[ \color{blue}{\frac{x \cdot \left(1 + y\right)}{z}} - x \]
      *-commutative [=>] 90.3	\[ \frac{\color{blue}{\left(1 + y\right) \cdot x}}{z} - x \]
      +-commutative [=>] 90.3	\[ \frac{\color{blue}{\left(y + 1\right)} \cdot x}{z} - x \]
      distribute-lft1-in [<=] 90.3	\[ \frac{\color{blue}{y \cdot x + x}}{z} - x \]
      *-commutative [=>] 90.3	\[ \frac{\color{blue}{x \cdot y} + x}{z} - x \]
      fma-def [=>] 90.3	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, x\right)}}{z} - x \]

   3. Taylor expanded in y around inf 90.3%

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

   4. Simplified 95.5%

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

      [Start] 90.3	\[ \frac{y \cdot x}{z} - x \]
      associate-/l* [=>] 95.5	\[ \color{blue}{\frac{y}{\frac{z}{x}}} - x \]

   5. Taylor expanded in x around 0 99.8%

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

   if -5.8e12 < z < 1.5999999999999999e32

   1. Initial program 99.6%

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

   if 1.5999999999999999e32 < z

   1. Initial program 70.7%

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

   2. Simplified 99.9%

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

      [Start] 70.7	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 99.9	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5800000000000:\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0034 \lor \neg \left(z \leq 1.7 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 2
Accuracy	93.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.02\right):\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 3
Accuracy	98.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.95 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 4
Accuracy	98.6%
Cost	712

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

Alternative 5
Accuracy	69.0%
Cost	588

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+31}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.07 \cdot 10^{-47}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Accuracy	69.0%
Cost	588

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+31}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 7
Accuracy	82.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+61} \lor \neg \left(y \leq 2.3 \cdot 10^{+59}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8
Accuracy	81.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 9
Accuracy	69.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-18}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 10
Accuracy	48.5%
Cost	128

\[-x \]

Alternative 11
Accuracy	3.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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))