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

Average Accuracy: 84.3% → 99.5%

Time: 7.9s

Precision: binary64

Cost: 841

Math

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

↓

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -2 \cdot 10^{+17} \lor \neg \left(z \leq 4 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

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 (or (<= z -2e+17) (not (<= z 4e-101)))
     (/ x (/ z t_0))
     (/ (* x t_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 t_0 = (y - z) + 1.0;
	double tmp;
	if ((z <= -2e+17) || !(z <= 4e-101)) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_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) :: t_0
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    if ((z <= (-2d+17)) .or. (.not. (z <= 4d-101))) then
        tmp = x / (z / t_0)
    else
        tmp = (x * t_0) / 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 t_0 = (y - z) + 1.0;
	double tmp;
	if ((z <= -2e+17) || !(z <= 4e-101)) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / z;
	}
	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 <= -2e+17) or not (z <= 4e-101):
		tmp = x / (z / t_0)
	else:
		tmp = (x * t_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)
	t_0 = Float64(Float64(y - z) + 1.0)
	tmp = 0.0
	if ((z <= -2e+17) || !(z <= 4e-101))
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x * t_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)
	t_0 = (y - z) + 1.0;
	tmp = 0.0;
	if ((z <= -2e+17) || ~((z <= 4e-101)))
		tmp = x / (z / t_0);
	else
		tmp = (x * t_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_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[z, -2e+17], N[Not[LessEqual[z, 4e-101]], $MachinePrecision]], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	84.3%
Target	99.2%
Herbie	99.5%

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

Derivation

1. Split input into 2 regimes

2. if z < -2e17 or 4.00000000000000021e-101 < z

   1. Initial program 76.8%

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

   2. Simplified 99.4%

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

      [Start] 76.8	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 99.4	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   if -2e17 < z < 4.00000000000000021e-101

   1. Initial program 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+17} \lor \neg \left(z \leq 4 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.7%
Cost	849

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+164}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+44} \lor \neg \left(y \leq 2.9 \cdot 10^{+71}\right) \land y \leq 4.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Accuracy	79.9%
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{z} - x\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+162}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Accuracy	80.7%
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{z} - x\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+161}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00037 \lor \neg \left(z \leq 2.9 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00037 \lor \neg \left(z \leq 2.3 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 6
Accuracy	65.3%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 7
Accuracy	83.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2700:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 8
Accuracy	70.2%
Cost	456

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

Alternative 9
Accuracy	48.6%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023135 
(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))