Average Error: 2.3 → 0.5
Time: 4.9s
Precision: binary64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, z, \mathsf{fma}\left(a \cdot \mathsf{fma}\left(z, b, t\right), 1, x\right)\right)\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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
 (let* ((t_1 (fma y z (fma (* a (fma z b t)) 1.0 x))))
   (if (<= a -6.8e-58)
     t_1
     (if (<= a 2e-116) (fma a t (fma z (fma a b y) x)) t_1))))
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 = fma(y, z, fma((a * fma(z, b, t)), 1.0, x));
	double tmp;
	if (a <= -6.8e-58) {
		tmp = t_1;
	} else if (a <= 2e-116) {
		tmp = fma(a, t, fma(z, fma(a, b, y), x));
	} else {
		tmp = t_1;
	}
	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)
	t_1 = fma(y, z, fma(Float64(a * fma(z, b, t)), 1.0, x))
	tmp = 0.0
	if (a <= -6.8e-58)
		tmp = t_1;
	elseif (a <= 2e-116)
		tmp = fma(a, t, fma(z, fma(a, b, y), x));
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(y * z + N[(N[(a * N[(z * b + t), $MachinePrecision]), $MachinePrecision] * 1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.8e-58], t$95$1, If[LessEqual[a, 2e-116], N[(a * t + N[(z * N[(a * b + y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
t_1 := \mathsf{fma}\left(y, z, \mathsf{fma}\left(a \cdot \mathsf{fma}\left(z, b, t\right), 1, x\right)\right)\\
\mathbf{if}\;a \leq -6.8 \cdot 10^{-58}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;a \leq 2 \cdot 10^{-116}:\\
\;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\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

Original2.3
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if a < -6.79999999999999947e-58 or 2e-116 < a

    1. Initial program 3.8

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

    2. Simplified0.9

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

    3. Applied egg-rr36.9

      \[\leadsto \mathsf{fma}\left(y, z, \color{blue}{\sqrt[3]{\mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)}^{2}}}\right) \]

    4. Applied egg-rr0.9

      \[\leadsto \mathsf{fma}\left(y, z, \color{blue}{\mathsf{fma}\left(a \cdot \mathsf{fma}\left(z, b, t\right), 1, x\right)}\right) \]

    if -6.79999999999999947e-58 < a < 2e-116

    1. Initial program 0.6

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

    2. Taylor expanded in x around 0 5.1

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

    3. Simplified0.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a \cdot \mathsf{fma}\left(z, b, t\right), 1, x\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a \cdot \mathsf{fma}\left(z, b, t\right), 1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))