Average Error: 2.1 → 1.2
Time: 4.8s
Precision: binary64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq 10^{+50}:\\ \;\;\;\;\left(\left(x + z \cdot y\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array} \]
(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
 (if (<= z 1e+50)
   (+ (+ (+ x (* z y)) (* t a)) (* (* z a) b))
   (fma a t (fma z (fma a b y) x))))
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 tmp;
	if (z <= 1e+50) {
		tmp = ((x + (z * y)) + (t * a)) + ((z * a) * b);
	} else {
		tmp = fma(a, t, fma(z, fma(a, b, y), x));
	}
	return tmp;
}

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)
	tmp = 0.0
	if (z <= 1e+50)
		tmp = Float64(Float64(Float64(x + Float64(z * y)) + Float64(t * a)) + Float64(Float64(z * a) * b));
	else
		tmp = fma(a, t, fma(z, fma(a, b, y), x));
	end
	return tmp
end

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_] := If[LessEqual[z, 1e+50], N[(N[(N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(a * t + N[(z * N[(a * b + y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;z \leq 10^{+50}:\\
\;\;\;\;\left(\left(x + z \cdot y\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original2.1
Target0.4
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < 1.0000000000000001e50

    1. Initial program 1.4

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

    if 1.0000000000000001e50 < z

    1. Initial program 6.5

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

    2. Taylor expanded in x around 0 8.9

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

    3. Simplified0.1

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+50}:\\ \;\;\;\;\left(\left(x + z \cdot y\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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)))