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

Average Accuracy: 84.3% → 99.3%

Time: 7.7s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -390000:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{elif}\;z \leq 10^{-111}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)))
   (if (<= z -390000.0)
     (- (/ x (/ z y)) x)
     (if (<= z 1e-111) (/ (* 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 = (y - z) + 1.0;
	double tmp;
	if (z <= -390000.0) {
		tmp = (x / (z / y)) - x;
	} else if (z <= 1e-111) {
		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 = (y - z) + 1.0d0
    if (z <= (-390000.0d0)) then
        tmp = (x / (z / y)) - x
    else if (z <= 1d-111) 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 = (y - z) + 1.0;
	double tmp;
	if (z <= -390000.0) {
		tmp = (x / (z / y)) - x;
	} else if (z <= 1e-111) {
		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 = (y - z) + 1.0
	tmp = 0
	if z <= -390000.0:
		tmp = (x / (z / y)) - x
	elif z <= 1e-111:
		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(Float64(y - z) + 1.0)
	tmp = 0.0
	if (z <= -390000.0)
		tmp = Float64(Float64(x / Float64(z / y)) - x);
	elseif (z <= 1e-111)
		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 = (y - z) + 1.0;
	tmp = 0.0;
	if (z <= -390000.0)
		tmp = (x / (z / y)) - x;
	elseif (z <= 1e-111)
		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[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[z, -390000.0], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 1e-111], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	84.3%
Target	99.3%
Herbie	99.3%

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

Derivation

1. Split input into 3 regimes

2. if z < -3.9e5

   1. Initial program 73.7%

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

   2. Simplified 92.2%

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

      [Start] 73.7	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      +-commutative [=>] 73.7	\[ \frac{x \cdot \color{blue}{\left(1 + \left(y - z\right)\right)}}{z} \]
      sub-neg [=>] 73.7	\[ \frac{x \cdot \left(1 + \color{blue}{\left(y + \left(-z\right)\right)}\right)}{z} \]
      +-commutative [=>] 73.7	\[ \frac{x \cdot \left(1 + \color{blue}{\left(\left(-z\right) + y\right)}\right)}{z} \]
      associate-+r+ [=>] 73.7	\[ \frac{x \cdot \color{blue}{\left(\left(1 + \left(-z\right)\right) + y\right)}}{z} \]
      unsub-neg [=>] 73.7	\[ \frac{x \cdot \left(\color{blue}{\left(1 - z\right)} + y\right)}{z} \]
      associate-+l- [=>] 73.7	\[ \frac{x \cdot \color{blue}{\left(1 - \left(z - y\right)\right)}}{z} \]
      distribute-lft-out-- [<=] 73.7	\[ \frac{\color{blue}{x \cdot 1 - x \cdot \left(z - y\right)}}{z} \]
      *-rgt-identity [=>] 73.7	\[ \frac{\color{blue}{x} - x \cdot \left(z - y\right)}{z} \]
      distribute-rgt-out-- [<=] 73.7	\[ \frac{x - \color{blue}{\left(z \cdot x - y \cdot x\right)}}{z} \]
      sub-neg [=>] 73.7	\[ \frac{x - \color{blue}{\left(z \cdot x + \left(-y \cdot x\right)\right)}}{z} \]
      +-commutative [=>] 73.7	\[ \frac{x - \color{blue}{\left(\left(-y \cdot x\right) + z \cdot x\right)}}{z} \]
      associate--r+ [=>] 73.7	\[ \frac{\color{blue}{\left(x - \left(-y \cdot x\right)\right) - z \cdot x}}{z} \]
      div-sub [=>] 73.7	\[ \color{blue}{\frac{x - \left(-y \cdot x\right)}{z} - \frac{z \cdot x}{z}} \]

   3. Taylor expanded in y around inf 91.7%

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

   4. Taylor expanded in y around 0 91.7%

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

   5. Simplified 99.4%

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

      [Start] 91.7	\[ \frac{y \cdot x}{z} + -1 \cdot x \]
      associate-*l/ [<=] 99.4	\[ \color{blue}{\frac{y}{z} \cdot x} + -1 \cdot x \]
      neg-mul-1 [<=] 99.4	\[ \frac{y}{z} \cdot x + \color{blue}{\left(-x\right)} \]
      sub-neg [<=] 99.4	\[ \color{blue}{\frac{y}{z} \cdot x - x} \]
      *-commutative [=>] 99.4	\[ \color{blue}{x \cdot \frac{y}{z}} - x \]
      associate-*r/ [=>] 91.7	\[ \color{blue}{\frac{x \cdot y}{z}} - x \]
      associate-/l* [=>] 99.4	\[ \color{blue}{\frac{x}{\frac{z}{y}}} - x \]

   if -3.9e5 < z < 1.00000000000000009e-111

   1. Initial program 99.8%

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

   if 1.00000000000000009e-111 < z

   1. Initial program 80.0%

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

   2. Simplified 98.7%

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

      [Start] 80.0	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 98.7	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -390000:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{elif}\;z \leq 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.7%
Cost	984

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -400000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 10^{-173}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.05:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -390000 \lor \neg \left(z \leq 10^{+16}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	841

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-71} \lor \neg \left(z \leq 1.6 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	76.9%
Cost	717

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+99} \lor \neg \left(y \leq 108000000000\right) \land y \leq 1.6 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 5
Accuracy	76.9%
Cost	717

\[\begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 85000000000 \lor \neg \left(y \leq 1.6 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 6
Accuracy	93.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -55 \lor \neg \left(y \leq 2.06 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 7
Accuracy	98.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.9 \lor \neg \left(z \leq 2.4 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 8
Accuracy	95.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -55:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;y \leq 2.06 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternative 9
Accuracy	95.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -55:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;y \leq 2.06 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{z}{1 - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternative 10
Accuracy	70.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -400000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.05:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 11
Accuracy	48.4%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023144 
(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))