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

Average Error: 9.9 → 0.2

Time: 5.5s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+80)
   (- (* (/ y z) x) x)
   (if (<= z 1e-34) (- (/ (fma x y x) z) x) (/ x (/ z (+ (- y z) 1.0))))))

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 tmp;
	if (z <= -1e+80) {
		tmp = ((y / z) * x) - x;
	} else if (z <= 1e-34) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = x / (z / ((y - z) + 1.0));
	}
	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)
	tmp = 0.0
	if (z <= -1e+80)
		tmp = Float64(Float64(Float64(y / z) * x) - x);
	elseif (z <= 1e-34)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) + 1.0)));
	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_] := If[LessEqual[z, -1e+80], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 1e-34], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.9
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 < -1e80

   1. Initial program 19.5

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

   2. Simplified 6.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      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): 0 points increase in error, 0 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): 16 points increase in error, 20 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, 0 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): 14 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z))) x): 20 points increase in error, 17 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z)))) x): 15 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): 0 points increase in error, 1 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): 19 points increase in error, 0 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, 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, 9 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): 8 points increase in error, 30 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, 9 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): 7 points increase in error, 3 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, 18 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, 55 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, 4 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): 51 points increase in error, 9 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, 53 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): 57 points increase in error, 1 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, 57 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): 57 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, 57 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))): 45 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)): 0 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): 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))): 0 points increase in error, 1 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)))): 64 points increase in error, 62 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 (/.f64 x z) (-.f64 1 (-.f64 z y)))): 1 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)): 62 points increase in error, 64 points decrease in error

   3. Taylor expanded in y around inf 6.1

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

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} - x \]
      Proof
      (*.f64 (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) z)): 55 points increase in error, 45 points decrease in error

   if -1e80 < z < 9.99999999999999928e-35

   1. Initial program 0.7

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

   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): 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): 0 points increase in error, 0 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): 16 points increase in error, 20 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, 0 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): 14 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z))) x): 20 points increase in error, 17 points decrease in error
      (-.f64 (+.f64 (*.f64 (/.f64 1 z) x) (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z)))) x): 15 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): 0 points increase in error, 1 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): 19 points increase in error, 0 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, 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, 9 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): 8 points increase in error, 30 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, 9 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): 7 points increase in error, 3 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, 18 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, 55 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, 4 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): 51 points increase in error, 9 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, 53 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): 57 points increase in error, 1 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, 57 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): 57 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, 57 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))): 45 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)): 0 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): 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))): 0 points increase in error, 1 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)))): 64 points increase in error, 62 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 (/.f64 x z) (-.f64 1 (-.f64 z y)))): 1 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)): 62 points increase in error, 64 points decrease in error

   if 9.99999999999999928e-35 < z

   1. Initial program 15.4

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]
      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)): 67 points increase in error, 15 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.0
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x\\ t_1 := \frac{x}{z} - x\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+234}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.8
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ t_1 := \frac{x}{z} - x\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+234}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.9
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{z} - x\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+234}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-34}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.2
Cost	840

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 6
Error	19.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 115000000:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 7
Error	4.5
Cost	712

\[\begin{array}{l} t_0 := x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{if}\;y \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	4.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \end{array} \]

Alternative 9
Error	3.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \end{array} \]

Alternative 10
Error	1.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.02:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \end{array} \]

Alternative 11
Error	20.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 12
Error	33.5
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022326 
(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))