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

Average Error: 10.2 → 1.0

Time: 7.7s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \left(y - z\right)\\ \mathbf{if}\;z \leq -2.626797904013124 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \frac{t_0}{z}\\ \mathbf{elif}\;z \leq 4.462524195544481 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \end{array} \]

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 -2.626797904013124e+50)
     (* x (/ t_0 z))
     (if (<= z 4.462524195544481e+25) (/ (* x t_0) z) (- (/ y (/ z x)) 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 t_0 = 1.0 + (y - z);
	double tmp;
	if (z <= -2.626797904013124e+50) {
		tmp = x * (t_0 / z);
	} else if (z <= 4.462524195544481e+25) {
		tmp = (x * t_0) / z;
	} else {
		tmp = (y / (z / x)) - 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y - z)
    if (z <= (-2.626797904013124d+50)) then
        tmp = x * (t_0 / z)
    else if (z <= 4.462524195544481d+25) then
        tmp = (x * t_0) / z
    else
        tmp = (y / (z / x)) - 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 t_0 = 1.0 + (y - z);
	double tmp;
	if (z <= -2.626797904013124e+50) {
		tmp = x * (t_0 / z);
	} else if (z <= 4.462524195544481e+25) {
		tmp = (x * t_0) / z;
	} else {
		tmp = (y / (z / x)) - x;
	}
	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 <= -2.626797904013124e+50:
		tmp = x * (t_0 / z)
	elif z <= 4.462524195544481e+25:
		tmp = (x * t_0) / z
	else:
		tmp = (y / (z / x)) - 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)
	t_0 = Float64(1.0 + Float64(y - z))
	tmp = 0.0
	if (z <= -2.626797904013124e+50)
		tmp = Float64(x * Float64(t_0 / z));
	elseif (z <= 4.462524195544481e+25)
		tmp = Float64(Float64(x * t_0) / z);
	else
		tmp = Float64(Float64(y / Float64(z / x)) - 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)
	t_0 = 1.0 + (y - z);
	tmp = 0.0;
	if (z <= -2.626797904013124e+50)
		tmp = x * (t_0 / z);
	elseif (z <= 4.462524195544481e+25)
		tmp = (x * t_0) / z;
	else
		tmp = (y / (z / x)) - 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_] := Block[{t$95$0 = N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.626797904013124e+50], N[(x * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.462524195544481e+25], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]]

Target

Original	10.2
Target	0.5
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if z < -2.6267979040131242e50

   1. Initial program 19.1

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

   2. Applied egg-rr 0.1

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

   if -2.6267979040131242e50 < z < 4.46252419554448063e25

   1. Initial program 0.4

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

   if 4.46252419554448063e25 < z

   1. Initial program 18.8

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

   2. Simplified 6.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      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): 0 points increase in error, 0 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): 1 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): 15 points increase in error, 14 points decrease in error
      (-.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (/.f64 x z) 1) (*.f64 (/.f64 x z) y))) x): 0 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): 14 points increase in error, 14 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z)))) x): 28 points increase in error, 11 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): 2 points increase in error, 2 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): 16 points increase in error, 5 points decrease in error
      (-.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z x) z)) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 68 points increase in error, 12 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 z x) (+.f64 (/.f64 1 z) (/.f64 y z))) z)) x): 14 points increase in error, 16 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): 1 points increase in error, 1 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): 1 points increase in error, 40 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): 0 points increase in error, 5 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): 5 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 (+.f64 x (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 (/.f64 1 z) (*.f64 z x))))) z) x): 5 points increase in error, 20 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): 9 points increase in error, 41 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 x) z)) z))) z) x): 4 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): 41 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, 41 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): 6 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, 3 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): 0 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, 0 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))): 30 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)): 4 points increase in error, 1 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, 5 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)))): 57 points increase in error, 39 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 (/.f64 x z) (-.f64 1 (-.f64 z y)))): 7 points increase in error, 1 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)): 41 points increase in error, 62 points decrease in error

   3. Taylor expanded in y around inf 6.2

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

   4. Taylor expanded in y around 0 6.2

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

   5. Simplified 2.9

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}} - x} \]
      Proof
      (-.f64 (/.f64 y (/.f64 z x)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) z)) x): 27 points increase in error, 20 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 y x) z) (neg.f64 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y x) z) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.626797904013124 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \frac{1 + \left(y - z\right)}{z}\\ \mathbf{elif}\;z \leq 4.462524195544481 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \end{array} \]

Alternatives

Alternative 1
Error	2.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq 4.462524195544481 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1864

\[\begin{array}{l} t_0 := 1 + \left(y - z\right)\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \end{array} \]

Alternative 3
Error	2.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -6.595237265751936 \cdot 10^{+27}:\\ \;\;\;\;\frac{x \cdot y}{z} - x\\ \mathbf{elif}\;z \leq 4.462524195544481 \cdot 10^{+25}:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \end{array} \]

Alternative 4
Error	0.2
Cost	840

\[\begin{array}{l} t_0 := \left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{+20}:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	2.9
Cost	712

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}} - x\\ \mathbf{if}\;y \leq -11520137663132634:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.229704819082429 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	2.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -11520137663132634:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;y \leq 3.229704819082429 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \end{array} \]

Alternative 7
Error	11.8
Cost	584

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.846686416611949 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	19.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -7016012.143991687:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.00019449687966690364:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9
Error	33.4
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022292 
(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))