Average Error: 6.3 → 0.9
Time: 5.2s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{if}\;a \leq -2 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 10^{-30}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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 y (/ (- t z) a) x)))
   (if (<= a -2e+69) t_1 (if (<= a 1e-30) (- x (/ (* y (- z t)) 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(y, ((t - z) / a), x);
	double tmp;
	if (a <= -2e+69) {
		tmp = t_1;
	} else if (a <= 1e-30) {
		tmp = x - ((y * (z - t)) / 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(y, Float64(Float64(t - z) / a), x)
	tmp = 0.0
	if (a <= -2e+69)
		tmp = t_1;
	elseif (a <= 1e-30)
		tmp = Float64(x - Float64(Float64(y * Float64(z - t)) / 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[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2e+69], t$95$1, If[LessEqual[a, 1e-30], N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
t_1 := \mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\
\mathbf{if}\;a \leq -2 \cdot 10^{+69}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;a \leq 10^{-30}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\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

Original6.3
Target0.7
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if a < -2.0000000000000001e69 or 1e-30 < a

    1. Initial program 9.9

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

    2. Simplified0.6

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

    if -2.0000000000000001e69 < a < 1e-30

    1. Initial program 1.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 10^{-30}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))