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

Average Error: 10.2 → 0.2

Time: 9.4s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5999249374451464 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;z \leq 14278328.396644196:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.5999249374451464e+40)
   (- (* x (/ y z)) x)
   (if (<= z 14278328.396644196)
     (* (/ x z) (+ (- y z) 1.0))
     (- (/ x (/ z y)) x))))

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 <= -3.5999249374451464e+40) {
		tmp = (x * (y / z)) - x;
	} else if (z <= 14278328.396644196) {
		tmp = (x / z) * ((y - z) + 1.0);
	} else {
		tmp = (x / (z / y)) - x;
	}
	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 <= (-3.5999249374451464d+40)) then
        tmp = (x * (y / z)) - x
    else if (z <= 14278328.396644196d0) then
        tmp = (x / z) * ((y - z) + 1.0d0)
    else
        tmp = (x / (z / y)) - x
    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 <= -3.5999249374451464e+40) {
		tmp = (x * (y / z)) - x;
	} else if (z <= 14278328.396644196) {
		tmp = (x / z) * ((y - z) + 1.0);
	} else {
		tmp = (x / (z / y)) - x;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if z <= -3.5999249374451464e+40:
		tmp = (x * (y / z)) - x
	elif z <= 14278328.396644196:
		tmp = (x / z) * ((y - z) + 1.0)
	else:
		tmp = (x / (z / y)) - x
	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 <= -3.5999249374451464e+40)
		tmp = Float64(Float64(x * Float64(y / z)) - x);
	elseif (z <= 14278328.396644196)
		tmp = Float64(Float64(x / z) * Float64(Float64(y - z) + 1.0));
	else
		tmp = Float64(Float64(x / Float64(z / y)) - x);
	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 <= -3.5999249374451464e+40)
		tmp = (x * (y / z)) - x;
	elseif (z <= 14278328.396644196)
		tmp = (x / z) * ((y - z) + 1.0);
	else
		tmp = (x / (z / y)) - x;
	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, -3.5999249374451464e+40], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 14278328.396644196], N[(N[(x / z), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]

Target

Original	10.2
Target	0.5
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 < -3.5999249374451464e40

   1. Initial program 18.0

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

   2. Simplified 6.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      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): 0 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 x y) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 y 1))) z) x): 2 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 y))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x z) (+.f64 1 y))) x): 17 points increase in error, 29 points decrease in error
      (-.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (/.f64 x z) 1) (*.f64 (/.f64 x z) y))) x): 2 points increase in error, 2 points decrease in error
      (-.f64 (+.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 x z)) (*.f64 (/.f64 x z) y)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) z) (*.f64 (/.f64 x z) y)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 z) x)) (*.f64 (/.f64 x z) y)) x): 15 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z))) x): 26 points increase in error, 16 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z)))) x): 25 points increase in error, 22 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y z) x))) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite=> distribute-rgt-out_binary64 (*.f64 x (+.f64 (/.f64 1 z) (/.f64 y z)))) x): 1 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 z z)) x) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 z (/.f64 z x))) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 13 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z x) z)) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 80 points increase in error, 10 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 z x) (+.f64 (/.f64 1 z) (/.f64 y z))) z)) x): 21 points increase in error, 15 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 1 z) (*.f64 z x)) (*.f64 (/.f64 y z) (*.f64 z x)))) z) x): 0 points increase in error, 2 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (/.f64 1 z) z) x)) (*.f64 (/.f64 y z) (*.f64 z x))) z) x): 7 points increase in error, 32 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 z (/.f64 1 z))) x) (*.f64 (/.f64 y z) (*.f64 z x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 (Rewrite=> rgt-mult-inverse_binary64 1) x) (*.f64 (/.f64 y z) (*.f64 z x))) z) x): 1 points increase in error, 11 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> *-lft-identity_binary64 x) (*.f64 (/.f64 y z) (*.f64 z x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)) z) (*.f64 z x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 1 z))) (*.f64 z x))) z) x): 9 points increase in error, 5 points decrease in error
      (-.f64 (/.f64 (+.f64 x (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 (/.f64 1 z) (*.f64 z x))))) z) x): 6 points increase in error, 27 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (/.f64 1 z) (Rewrite=> *-commutative_binary64 (*.f64 x z))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 z) x) z)))) z) x): 10 points increase in error, 58 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 x) z)) z))) z) x): 1 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 x) z) z))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite=> *-commutative_binary64 (*.f64 z (/.f64 x z))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z x) z)))) z) x): 58 points increase in error, 8 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 z z) x)))) z) x): 0 points increase in error, 58 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (Rewrite=> *-inverses_binary64 1) x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite=> *-lft-identity_binary64 x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (Rewrite<= *-commutative_binary64 (*.f64 x y))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 x (*.f64 x y))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 x) (neg.f64 (*.f64 x y))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (neg.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))) (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (neg.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 z z)) x)) (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 z (/.f64 z x)))) (neg.f64 (*.f64 x y)))) z) x): 4 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 z) (/.f64 z x))) (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 z) x) z)) (neg.f64 (*.f64 x y)))) z) x): 52 points increase in error, 4 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 z) (/.f64 x z))) (neg.f64 (*.f64 x y)))) z) x): 1 points increase in error, 52 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 (*.f64 (neg.f64 z) (/.f64 x z))) (neg.f64 (neg.f64 (*.f64 x y))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (neg.f64 z)) (/.f64 x z))) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 (Rewrite=> remove-double-neg_binary64 z) (/.f64 x z)) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z x) z)) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 52 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 z z) x)) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 52 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 (Rewrite=> *-inverses_binary64 1) x) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> *-lft-identity_binary64 x) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (-.f64 x (neg.f64 (*.f64 x y))) z) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (-.f64 x (neg.f64 (*.f64 x y))) z) (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 z z)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (-.f64 x (neg.f64 (*.f64 x y))) z) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 z x) z))): 39 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (-.f64 x (neg.f64 (*.f64 x y))) (*.f64 z x)) z)): 2 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 x (+.f64 (neg.f64 (*.f64 x y)) (*.f64 z x)))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 z x) (neg.f64 (*.f64 x y))))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 z x) (*.f64 x y)))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 x z)) (*.f64 x y))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (Rewrite=> distribute-lft-out--_binary64 (*.f64 x (-.f64 z y)))) z): 1 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x z) (/.f64 (*.f64 x (-.f64 z y)) z))): 1 points increase in error, 3 points decrease in error
      (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 x z) 1)) (/.f64 (*.f64 x (-.f64 z y)) z)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (/.f64 x z) 1) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x z) (-.f64 z y)))): 54 points increase in error, 58 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 (/.f64 x z) (-.f64 1 (-.f64 z y)))): 2 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 x z) (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 1 z) y))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z))) y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 (neg.f64 z) y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 y z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 y z) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)): 63 points increase in error, 55 points decrease in error

   3. Taylor expanded in y around 0 6.3

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

   4. Taylor expanded in y around inf 6.3

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

   5. Simplified 0.1

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} - x \]
      Proof
      (*.f64 x (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) z)): 53 points increase in error, 51 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) z): 0 points increase in error, 0 points decrease in error

   if -3.5999249374451464e40 < z < 14278328.3966441955

   1. Initial program 0.4

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

   2. Applied egg-rr 0.3

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

   if 14278328.3966441955 < z

   1. Initial program 17.8

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

   2. Simplified 5.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      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): 0 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 x y) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 y 1))) z) x): 2 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 y))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x z) (+.f64 1 y))) x): 17 points increase in error, 29 points decrease in error
      (-.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (/.f64 x z) 1) (*.f64 (/.f64 x z) y))) x): 2 points increase in error, 2 points decrease in error
      (-.f64 (+.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 x z)) (*.f64 (/.f64 x z) y)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) z) (*.f64 (/.f64 x z) y)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 z) x)) (*.f64 (/.f64 x z) y)) x): 15 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z))) x): 26 points increase in error, 16 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z)))) x): 25 points increase in error, 22 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y z) x))) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite=> distribute-rgt-out_binary64 (*.f64 x (+.f64 (/.f64 1 z) (/.f64 y z)))) x): 1 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 z z)) x) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 z (/.f64 z x))) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 13 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z x) z)) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 80 points increase in error, 10 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 z x) (+.f64 (/.f64 1 z) (/.f64 y z))) z)) x): 21 points increase in error, 15 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 1 z) (*.f64 z x)) (*.f64 (/.f64 y z) (*.f64 z x)))) z) x): 0 points increase in error, 2 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (/.f64 1 z) z) x)) (*.f64 (/.f64 y z) (*.f64 z x))) z) x): 7 points increase in error, 32 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 z (/.f64 1 z))) x) (*.f64 (/.f64 y z) (*.f64 z x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 (Rewrite=> rgt-mult-inverse_binary64 1) x) (*.f64 (/.f64 y z) (*.f64 z x))) z) x): 1 points increase in error, 11 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> *-lft-identity_binary64 x) (*.f64 (/.f64 y z) (*.f64 z x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)) z) (*.f64 z x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 1 z))) (*.f64 z x))) z) x): 9 points increase in error, 5 points decrease in error
      (-.f64 (/.f64 (+.f64 x (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 (/.f64 1 z) (*.f64 z x))))) z) x): 6 points increase in error, 27 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (/.f64 1 z) (Rewrite=> *-commutative_binary64 (*.f64 x z))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 z) x) z)))) z) x): 10 points increase in error, 58 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 x) z)) z))) z) x): 1 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 x) z) z))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite=> *-commutative_binary64 (*.f64 z (/.f64 x z))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z x) z)))) z) x): 58 points increase in error, 8 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 z z) x)))) z) x): 0 points increase in error, 58 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (Rewrite=> *-inverses_binary64 1) x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite=> *-lft-identity_binary64 x))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 x (Rewrite<= *-commutative_binary64 (*.f64 x y))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 x (*.f64 x y))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 x) (neg.f64 (*.f64 x y))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (neg.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))) (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (neg.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 z z)) x)) (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 z (/.f64 z x)))) (neg.f64 (*.f64 x y)))) z) x): 4 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 z) (/.f64 z x))) (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 z) x) z)) (neg.f64 (*.f64 x y)))) z) x): 52 points increase in error, 4 points decrease in error
      (-.f64 (/.f64 (neg.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 z) (/.f64 x z))) (neg.f64 (*.f64 x y)))) z) x): 1 points increase in error, 52 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 (*.f64 (neg.f64 z) (/.f64 x z))) (neg.f64 (neg.f64 (*.f64 x y))))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (neg.f64 z)) (/.f64 x z))) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 (Rewrite=> remove-double-neg_binary64 z) (/.f64 x z)) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z x) z)) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 52 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 z z) x)) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 52 points decrease in error
      (-.f64 (/.f64 (+.f64 (*.f64 (Rewrite=> *-inverses_binary64 1) x) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (+.f64 (Rewrite=> *-lft-identity_binary64 x) (neg.f64 (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (neg.f64 (*.f64 x y)))) z) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (-.f64 x (neg.f64 (*.f64 x y))) z) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (-.f64 x (neg.f64 (*.f64 x y))) z) (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 z z)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (-.f64 x (neg.f64 (*.f64 x y))) z) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 z x) z))): 39 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (-.f64 x (neg.f64 (*.f64 x y))) (*.f64 z x)) z)): 2 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 x (+.f64 (neg.f64 (*.f64 x y)) (*.f64 z x)))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 z x) (neg.f64 (*.f64 x y))))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 z x) (*.f64 x y)))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 x z)) (*.f64 x y))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (Rewrite=> distribute-lft-out--_binary64 (*.f64 x (-.f64 z y)))) z): 1 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x z) (/.f64 (*.f64 x (-.f64 z y)) z))): 1 points increase in error, 3 points decrease in error
      (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 x z) 1)) (/.f64 (*.f64 x (-.f64 z y)) z)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (/.f64 x z) 1) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x z) (-.f64 z y)))): 54 points increase in error, 58 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 (/.f64 x z) (-.f64 1 (-.f64 z y)))): 2 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 x z) (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 1 z) y))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z))) y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 (neg.f64 z) y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (+.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 y z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x z) (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 y z) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)): 63 points increase in error, 55 points decrease in error

   3. Taylor expanded in y around 0 5.7

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

   4. Taylor expanded in y around inf 6.0

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

   5. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} - x \]
      Proof
      (*.f64 x (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) z)): 53 points increase in error, 51 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) z): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in x around 0 0.3

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

   7. Simplified 0.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}} - x} \]
      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): 28 points increase in error, 38 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z))) x): 31 points increase in error, 31 points decrease in error
      (-.f64 (*.f64 x (/.f64 y z)) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 x (-.f64 (/.f64 y z) 1))): 1 points increase in error, 3 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 y z) 1) x)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5999249374451464 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;z \leq 14278328.396644196:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternatives

Alternative 1
Error	21.2
Cost	984

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -8262446372518119000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-259}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-271}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.2579608320672296 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Error	0.2
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-48}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.4
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -8262446372518119000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 2.2579608320672296 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	9.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6780228454531155 \cdot 10^{+44}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 5
Error	1.5
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}} - x\\ \mathbf{if}\;z \leq -8262446372518119000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.2579608320672296 \cdot 10^{-10}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -8262446372518119000:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;z \leq 2.2579608320672296 \cdot 10^{-10}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternative 7
Error	1.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -8262446372518119000:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;z \leq 2.2579608320672296 \cdot 10^{-10}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternative 8
Error	12.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -9.9 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 9
Error	11.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -9.9 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 10
Error	20.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -8262446372518119000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2.2579608320672296 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 11
Error	33.5
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022312 
(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))