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

Average Error: 10.6 → 0.3

Time: 7.0s

Precision: binary64

Cost: 8136

Math

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

↓

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)) (t_1 (/ (* x t_0) z)))
   (if (<= t_1 (- INFINITY))
     (- (/ y (/ z x)) x)
     (if (<= t_1 2e+276) (- (/ (fma x y x) z) x) (/ x (/ z 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 = (y - z) + 1.0;
	double t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / (z / x)) - x;
	} else if (t_1 <= 2e+276) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = x / (z / 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(Float64(y - z) + 1.0)
	t_1 = Float64(Float64(x * t_0) / z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / Float64(z / x)) - x);
	elseif (t_1 <= 2e+276)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = Float64(x / Float64(z / 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[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[t$95$1, 2e+276], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	10.6
Target	0.5
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 23.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      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): 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, 1 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): 6 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 1 y) x)) z) x): 2 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 1 y) z) x)) x): 0 points increase in error, 8 points decrease in error
      (-.f64 (*.f64 (/.f64 (+.f64 1 y) z) x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))): 3 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) 1))): 0 points increase in error, 3 points decrease in error
      (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) (Rewrite<= *-inverses_binary64 (/.f64 z z)))): 7 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 1 y) z) z))): 0 points increase in error, 7 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 1 (-.f64 y z))) z)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 y z) 1)) z)): 1 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)): 5 points increase in error, 1 points decrease in error

   3. Taylor expanded in y around inf 23.3

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} - x \]
      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): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 2.0000000000000001e276

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
      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): 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, 1 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): 6 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 1 y) x)) z) x): 2 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 1 y) z) x)) x): 0 points increase in error, 8 points decrease in error
      (-.f64 (*.f64 (/.f64 (+.f64 1 y) z) x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))): 3 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) 1))): 0 points increase in error, 3 points decrease in error
      (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) (Rewrite<= *-inverses_binary64 (/.f64 z z)))): 7 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 1 y) z) z))): 0 points increase in error, 7 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 1 (-.f64 y z))) z)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 y z) 1)) z)): 1 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)): 5 points increase in error, 1 points decrease in error

   if 2.0000000000000001e276 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 54.9

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

   2. Simplified 2.6

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]
      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, 1 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): 6 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 1 y) x)) z) x): 2 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 1 y) z) x)) x): 0 points increase in error, 8 points decrease in error
      (-.f64 (*.f64 (/.f64 (+.f64 1 y) z) x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))): 3 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) 1))): 0 points increase in error, 3 points decrease in error
      (*.f64 x (-.f64 (/.f64 (+.f64 1 y) z) (Rewrite<= *-inverses_binary64 (/.f64 z z)))): 7 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 1 y) z) z))): 0 points increase in error, 7 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 1 (-.f64 y z))) z)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 y z) 1)) z)): 1 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)): 5 points increase in error, 1 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.4
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+34}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -135:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+103}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.2
Cost	1112

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+34}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -135:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+106}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	13.4
Cost	1112

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+34}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -135:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Error	20.2
Cost	848

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+16}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 320000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 5
Error	20.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-106}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3200000000000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Error	0.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-104} \lor \neg \left(z \leq 2 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]

Alternative 7
Error	0.2
Cost	841

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+15} \lor \neg \left(z \leq 1.55 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

Alternative 8
Error	2.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -7.4 \lor \neg \left(y \leq 2.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 9
Error	19.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-8}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 10
Error	32.8
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022340 
(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))