Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 1.9 → 0.4

Time: 6.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := a \cdot t + \left(z \cdot \left(a \cdot b + y\right) + x\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ (* a t) (+ (* z (+ (* a b) y)) x))))
   (if (<= z -1e+145)
     t_1
     (if (<= z 1e-10) (fma a (fma b z t) (fma z y x)) t_1))))

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 = (a * t) + ((z * ((a * b) + y)) + x);
	double tmp;
	if (z <= -1e+145) {
		tmp = t_1;
	} else if (z <= 1e-10) {
		tmp = fma(a, fma(b, z, t), fma(z, y, x));
	} else {
		tmp = t_1;
	}
	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(a * t) + Float64(Float64(z * Float64(Float64(a * b) + y)) + x))
	tmp = 0.0
	if (z <= -1e+145)
		tmp = t_1;
	elseif (z <= 1e-10)
		tmp = fma(a, fma(b, z, t), fma(z, y, x));
	else
		tmp = t_1;
	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[(a * t), $MachinePrecision] + N[(N[(z * N[(N[(a * b), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+145], t$95$1, If[LessEqual[z, 1e-10], N[(a * N[(b * z + t), $MachinePrecision] + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	1.9
Target	0.4
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999999e144 or 1.00000000000000004e-10 < z

   1. Initial program 5.6

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

   2. Simplified 8.8

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

   3. Taylor expanded in a around 0 8.8

      \[\leadsto \mathsf{fma}\left(y, z, \color{blue}{\left(t + b \cdot z\right) \cdot a + x}\right) \]

   4. Taylor expanded in y around 0 8.8

      \[\leadsto \color{blue}{y \cdot z + \left(a \cdot \left(z \cdot b + t\right) + x\right)} \]

   5. Simplified 8.8

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

   6. Taylor expanded in z around inf 0.1

      \[\leadsto \color{blue}{a \cdot t + \left(z \cdot \left(a \cdot b + y\right) + x\right)} \]

   if -9.9999999999999999e144 < z < 1.00000000000000004e-10

   1. Initial program 0.6

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

   2. Simplified 0.5

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

   3. Taylor expanded in a around 0 0.5

      \[\leadsto \mathsf{fma}\left(y, z, \color{blue}{\left(t + b \cdot z\right) \cdot a + x}\right) \]

   4. Taylor expanded in y around 0 0.5

      \[\leadsto \color{blue}{y \cdot z + \left(a \cdot \left(z \cdot b + t\right) + x\right)} \]

   5. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), \mathsf{fma}\left(z, y, x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+145}:\\ \;\;\;\;a \cdot t + \left(z \cdot \left(a \cdot b + y\right) + x\right)\\ \mathbf{elif}\;z \leq 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t + \left(z \cdot \left(a \cdot b + y\right) + x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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)))