Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.4% → 97.6%

Time: 11.2s

Precision: binary64

Cost: 20676

• 32.3% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]

↓

\[\begin{array}{l} t_1 := \left(t \cdot a + \left(x + y \cdot z\right)\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{if}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (* t a) (+ x (* y z))) (* (* z a) b))))
   (if (<= t_1 1e+307) t_1 (fma y z (fma a (fma z b t) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + (y * z)) + (t * a)) + ((a * z) * b);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t * a) + (x + (y * z))) + ((z * a) * b);
	double tmp;
	if (t_1 <= 1e+307) {
		tmp = t_1;
	} else {
		tmp = fma(y, z, fma(a, fma(z, b, t), x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t * a) + Float64(x + Float64(y * z))) + Float64(Float64(z * a) * b))
	tmp = 0.0
	if (t_1 <= 1e+307)
		tmp = t_1;
	else
		tmp = fma(y, z, fma(a, fma(z, b, t), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t * a), $MachinePrecision] + N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+307], t$95$1, N[(y * z + N[(a * N[(z * b + t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	92.4%
Target	97.6%
Herbie	97.6%

\[\begin{array}{l} \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < 9.99999999999999986e306

   1. Initial program 99.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]

   if 9.99999999999999986e306 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 66.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]

   2. Simplified 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)} \]
      Step-by-step derivation

      [Start] 66.0%	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 66.0%	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 66.0%	\[ \color{blue}{\left(y \cdot z + x\right)} + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right) \]
      associate-+l+ [=>] 66.0%	\[ \color{blue}{y \cdot z + \left(x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]
      fma-def [=>] 70.9%	\[ \color{blue}{\mathsf{fma}\left(y, z, x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]
      +-commutative [=>] 70.9%	\[ \mathsf{fma}\left(y, z, \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + x}\right) \]
      *-commutative [=>] 70.9%	\[ \mathsf{fma}\left(y, z, \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + x\right) \]
      associate-*l* [=>] 85.4%	\[ \mathsf{fma}\left(y, z, \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + x\right) \]
      distribute-lft-out [=>] 97.6%	\[ \mathsf{fma}\left(y, z, \color{blue}{a \cdot \left(t + z \cdot b\right)} + x\right) \]
      fma-def [=>] 97.6%	\[ \mathsf{fma}\left(y, z, \color{blue}{\mathsf{fma}\left(a, t + z \cdot b, x\right)}\right) \]
      +-commutative [=>] 97.6%	\[ \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \color{blue}{z \cdot b + t}, x\right)\right) \]
      fma-def [=>] 97.6%	\[ \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(z, b, t\right)}, x\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t \cdot a + \left(x + y \cdot z\right)\right) + \left(z \cdot a\right) \cdot b \leq 10^{+307}:\\ \;\;\;\;\left(t \cdot a + \left(x + y \cdot z\right)\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	20676

\[\begin{array}{l} t_1 := \left(t \cdot a + \left(x + y \cdot z\right)\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{if}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	96.7%
Cost	1988

\[\begin{array}{l} t_1 := \left(t \cdot a + \left(x + y \cdot z\right)\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 3
Accuracy	35.6%
Cost	1644

\[\begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{+239}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+138}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.4 \cdot 10^{-256}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-83}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+145}:\\ \;\;\;\;t \cdot a\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4
Accuracy	37.4%
Cost	1376

\[\begin{array}{l} t_1 := z \cdot \left(a \cdot b\right)\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+189}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq -1.78 \cdot 10^{-273}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+76}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+89}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	92.8%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+170}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot a + a \cdot \left(z \cdot b\right)\right) + \left(x + y \cdot z\right)\\ \end{array} \]

Alternative 6
Accuracy	58.8%
Cost	980

\[\begin{array}{l} t_1 := x + t \cdot a\\ t_2 := z \cdot \left(a \cdot b\right)\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+190}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{+126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+130}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	39.4%
Cost	852

\[\begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+31}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq -1.08 \cdot 10^{-260}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1.85 \cdot 10^{-297}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]

Alternative 8
Accuracy	80.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -7200000000000 \lor \neg \left(a \leq 1.26 \cdot 10^{-31}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 9
Accuracy	86.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+89} \lor \neg \left(a \leq 7.5 \cdot 10^{-23}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + t \cdot a\right) + y \cdot z\\ \end{array} \]

Alternative 10
Accuracy	61.0%
Cost	716

\[\begin{array}{l} t_1 := x + t \cdot a\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.75 \cdot 10^{-31}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \]

Alternative 11
Accuracy	73.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+15} \lor \neg \left(a \leq 1.52 \cdot 10^{-15}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 12
Accuracy	39.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+30}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]

Alternative 13
Accuracy	26.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000.0) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))