Average Error: 6.2 → 1.5
Time: 4.8s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{if}\;z - t \leq -1.8034715821452683 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z - t \leq 1.2397713062370286 \cdot 10^{+28}:\\ \;\;\;\;x + y \cdot \left(\left(z - t\right) \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- z t) (/ y a) x)))
   (if (<= (- z t) -1.8034715821452683e-59)
     t_1
     (if (<= (- z t) 1.2397713062370286e+28)
       (+ x (* y (* (- z t) (/ 1.0 a))))
       t_1))))
double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / a);
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z - t), (y / a), x);
	double tmp;
	if ((z - t) <= -1.8034715821452683e-59) {
		tmp = t_1;
	} else if ((z - t) <= 1.2397713062370286e+28) {
		tmp = x + (y * ((z - t) * (1.0 / a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y * Float64(z - t)) / a))
end
↓
function code(x, y, z, t, a)
	t_1 = fma(Float64(z - t), Float64(y / a), x)
	tmp = 0.0
	if (Float64(z - t) <= -1.8034715821452683e-59)
		tmp = t_1;
	elseif (Float64(z - t) <= 1.2397713062370286e+28)
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) * Float64(1.0 / a))));
	else
		tmp = t_1;
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[(z - t), $MachinePrecision], -1.8034715821452683e-59], t$95$1, If[LessEqual[N[(z - t), $MachinePrecision], 1.2397713062370286e+28], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
t_1 := \mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\
\mathbf{if}\;z - t \leq -1.8034715821452683 \cdot 10^{-59}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;z - t \leq 1.2397713062370286 \cdot 10^{+28}:\\
\;\;\;\;x + y \cdot \left(\left(z - t\right) \cdot \frac{1}{a}\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

Target

Original6.2
Target0.7
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 z t) < -1.8034715821452683e-59 or 1.2397713062370286e28 < (-.f64 z t)

    1. Initial program 8.3

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

    2. Simplified7.9

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

    3. Taylor expanded in y around 0 8.3

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

    4. Simplified1.7

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

    if -1.8034715821452683e-59 < (-.f64 z t) < 1.2397713062370286e28

    1. Initial program 0.8

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

    2. Applied add-cube-cbrt_binary641.2

      \[\leadsto x + \frac{y \cdot \left(z - t\right)}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} \]

    3. Applied times-frac_binary642.4

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}} \]

    4. Applied div-inv_binary642.4

      \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)} \cdot \frac{z - t}{\sqrt[3]{a}} \]

    5. Applied associate-*l*_binary641.3

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}\right)} \]

    6. Simplified1.0

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{a} \cdot \left(z - t\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z - t \leq -1.8034715821452683 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;z - t \leq 1.2397713062370286 \cdot 10^{+28}:\\ \;\;\;\;x + y \cdot \left(\left(z - t\right) \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))