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

Average Error: 10.3 → 0.2

Time: 7.4s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x + x \cdot y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -4e+63)
   (- (/ x (/ z y)) x)
   (if (<= z 5.5e-30) (- (/ (+ x (* x y)) z) x) (* x (/ (- y (+ z -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 <= -4e+63) {
		tmp = (x / (z / y)) - x;
	} else if (z <= 5.5e-30) {
		tmp = ((x + (x * y)) / z) - x;
	} else {
		tmp = x * ((y - (z + -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 <= (-4d+63)) then
        tmp = (x / (z / y)) - x
    else if (z <= 5.5d-30) then
        tmp = ((x + (x * y)) / z) - x
    else
        tmp = x * ((y - (z + (-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 <= -4e+63) {
		tmp = (x / (z / y)) - x;
	} else if (z <= 5.5e-30) {
		tmp = ((x + (x * y)) / z) - x;
	} else {
		tmp = x * ((y - (z + -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 <= -4e+63:
		tmp = (x / (z / y)) - x
	elif z <= 5.5e-30:
		tmp = ((x + (x * y)) / z) - x
	else:
		tmp = x * ((y - (z + -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 <= -4e+63)
		tmp = Float64(Float64(x / Float64(z / y)) - x);
	elseif (z <= 5.5e-30)
		tmp = Float64(Float64(Float64(x + Float64(x * y)) / z) - x);
	else
		tmp = Float64(x * Float64(Float64(y - Float64(z + -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 <= -4e+63)
		tmp = (x / (z / y)) - x;
	elseif (z <= 5.5e-30)
		tmp = ((x + (x * y)) / z) - x;
	else
		tmp = x * ((y - (z + -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[LessEqual[z, -4e+63], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 5.5e-30], N[(N[(N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(x * N[(N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	10.3
Target	0.4
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if z < -4.00000000000000023e63

   1. Initial program 20.3

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

   2. Simplified 7.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      Proof
      (-.f64 (/.f64 (fma.f64 x y x) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) x)) z) x): 1 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 x y) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))) z) x): 6 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 y 1))) z) x): 1 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 y))) z) x): 0 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 1 y) x)) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 1 y) z) x)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (/.f64 (+.f64 1 y) z) x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))): 9 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) 1))): 0 points increase in error, 9 points decrease in error
      (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) (Rewrite<= *-inverses_binary64 (/.f64 z z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 1 y) z) z))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 1 (-.f64 y z))) z)): 6 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 y z) 1)) z)): 0 points increase in error, 6 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 7.2

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

   4. Simplified 2.3

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} - x \]
      Proof
      (-.f64 (*.f64 y (/.f64 x z)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 y (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 x))) z)) x): 0 points increase in error, 7 points decrease in error
      (-.f64 (*.f64 y (/.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) z)) x): 4 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 y (/.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 -1) x)) z)) x): 0 points increase in error, 4 points decrease in error
      (-.f64 (*.f64 y (/.f64 (*.f64 (Rewrite=> metadata-eval 1) x) z)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 y (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 z) x))) x): 11 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 1 z) x) y)) x): 0 points increase in error, 11 points decrease in error
      (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 z) (*.f64 x y))) x): 11 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (/.f64 1 z) (Rewrite<= *-commutative_binary64 (*.f64 y x))) x): 0 points increase in error, 11 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 (*.f64 y x)) z)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 (*.f64 y x)) z) x): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in y around 0 7.2

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

   6. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} - x \]
      Proof
      (-.f64 (/.f64 x (/.f64 z y)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x y) z)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) z) x): 3 points increase in error, 0 points decrease in error

   if -4.00000000000000023e63 < z < 5.49999999999999976e-30

   1. Initial program 0.8

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

   2. Simplified 0.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      Proof
      (-.f64 (/.f64 (fma.f64 x y x) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) x)) z) x): 1 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 x y) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))) z) x): 6 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 y 1))) z) x): 1 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 y))) z) x): 0 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 1 y) x)) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 1 y) z) x)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (/.f64 (+.f64 1 y) z) x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))): 9 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) 1))): 0 points increase in error, 9 points decrease in error
      (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) (Rewrite<= *-inverses_binary64 (/.f64 z z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 1 y) z) z))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 1 (-.f64 y z))) z)): 6 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 y z) 1)) z)): 0 points increase in error, 6 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in z around 0 0.4

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

   if 5.49999999999999976e-30 < z

   1. Initial program 15.0

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

   2. Applied egg-rr 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x + x \cdot y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-17} \lor \neg \left(z \leq 5.7 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 2
Error	19.0
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 10500000000000:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3
Error	4.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.7 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 4
Error	2.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.7 \cdot 10^{-13}\right):\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 5
Error	2.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.7 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 6
Error	1.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3300000:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \]

Alternative 7
Error	11.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+125} \lor \neg \left(y \leq 3800000000000\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8
Error	11.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+125} \lor \neg \left(y \leq 3800000000000\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 9
Error	19.0
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 10
Error	33.0
Cost	128

\[-x \]

Alternative 11
Error	62.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(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))