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

Average Error: 9.9 → 0.5

Time: 5.2s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (/ z (+ (- y z) 1.0)))))
   (if (<= x -1e+98) t_0 (if (<= x 1.6e-117) (- (/ (fma x y x) z) x) 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 = x / (z / ((y - z) + 1.0));
	double tmp;
	if (x <= -1e+98) {
		tmp = t_0;
	} else if (x <= 1.6e-117) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = 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(x / Float64(z / Float64(Float64(y - z) + 1.0)))
	tmp = 0.0
	if (x <= -1e+98)
		tmp = t_0;
	elseif (x <= 1.6e-117)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = t_0;
	end
	return 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[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+98], t$95$0, If[LessEqual[x, 1.6e-117], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

Target

Original	9.9
Target	0.3
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999998e97 or 1.59999999999999998e-117 < x

   1. Initial program 21.3

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

   2. Simplified 0.6

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]
      Proof
      (/.f64 x (/.f64 z (+.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, 29 points decrease in error

   if -9.99999999999999998e97 < x < 1.59999999999999998e-117

   1. Initial program 1.4

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      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): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 y 1))) z) x): 0 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, 21 points decrease in error
      (-.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (/.f64 x z) 1) (*.f64 (/.f64 x z) y))) x): 1 points increase in error, 1 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): 22 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): 22 points increase in error, 15 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, 19 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, 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): 10 points increase in error, 4 points decrease in error
      (-.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z x) z)) (+.f64 (/.f64 1 z) (/.f64 y z))) x): 79 points increase in error, 8 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 z x) (+.f64 (/.f64 1 z) (/.f64 y z))) z)) x): 17 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): 7 points increase in error, 33 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, 13 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, 2 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, 23 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): 12 points increase in error, 59 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, 11 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): 59 points increase in error, 7 points decrease in error
      (-.f64 (/.f64 (+.f64 x (*.f64 y (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 z z) x)))) z) x): 1 points increase in error, 59 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): 59 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): 1 points increase in error, 59 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): 59 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, 59 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))): 40 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, 0 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): 0 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)))): 60 points increase in error, 60 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 (/.f64 x z) (-.f64 1 (-.f64 z y)))): 3 points increase in error, 0 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)): 61 points increase in error, 61 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.3
Cost	1244

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+58}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Error	11.9
Cost	848

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+79}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	11.9
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+77}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	12.0
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{+76}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 5
Error	0.2
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.4
Cost	840

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ t_1 := \frac{x}{\frac{z}{t_0}}\\ \mathbf{if}\;x \leq -6 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	3.8
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z} - x\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.00098:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	3.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;y \leq 0.00098:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternative 9
Error	3.9
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z} - x\\ \mathbf{if}\;z \leq -0.98:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7800000:\\ \;\;\;\;\frac{x}{\frac{z}{y + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	1.0
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z} - x\\ \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	19.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7800000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 12
Error	32.6
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022329 
(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))