?

Average Error: 6.0 → 0.8
Time: 12.0s
Precision: binary64
Cost: 7113

?

\[x + \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{-73} \lor \neg \left(a \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.6e-73) (not (<= a 5e-16)))
   (fma y (/ (- z t) a) x)
   (+ x (/ (* y (- z t)) a))))
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 tmp;
	if ((a <= -7.6e-73) || !(a <= 5e-16)) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = x + ((y * (z - t)) / a);
	}
	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)
	tmp = 0.0
	if ((a <= -7.6e-73) || !(a <= 5e-16))
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	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_] := If[Or[LessEqual[a, -7.6e-73], N[Not[LessEqual[a, 5e-16]], $MachinePrecision]], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \leq -7.6 \cdot 10^{-73} \lor \neg \left(a \leq 5 \cdot 10^{-16}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\


\end{array}

Error?

Target

Original6.0
Target0.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation?

  1. Split input into 2 regimes

  2. if a < -7.6000000000000005e-73 or 5.0000000000000004e-16 < a

    1. Initial program 8.3

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

    2. Simplified0.8

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

      [Start]8.3

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

      +-commutative [=>]8.3

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

      associate-*r/ [<=]0.8

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

      fma-def [=>]0.8

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

    if -7.6000000000000005e-73 < a < 5.0000000000000004e-16

    1. Initial program 0.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{-73} \lor \neg \left(a \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error19.0
Cost1373
\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a}\\ t_2 := x + \frac{y}{\frac{a}{z}}\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+84} \lor \neg \left(a \leq 3.3 \cdot 10^{+92}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\frac{y}{\frac{a}{t}}\\ \end{array} \]
Alternative 2
Error1.4
Cost1097
\[\begin{array}{l} \mathbf{if}\;z - t \leq -5 \cdot 10^{+58} \lor \neg \left(z - t \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]
Alternative 3
Error29.7
Cost980
\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1300000000000:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-136}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 4
Error29.7
Cost980
\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1300000000000:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-160}:\\ \;\;\;\;-y \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 5
Error22.1
Cost977
\[\begin{array}{l} \mathbf{if}\;x \leq -1.76 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+19} \lor \neg \left(x \leq -4.2 \cdot 10^{-133}\right) \land x \leq 1.72 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 6
Error19.3
Cost977
\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1450000000 \lor \neg \left(x \leq -3.1 \cdot 10^{-38}\right) \land x \leq 1.25 \cdot 10^{-24}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 7
Error30.9
Cost850
\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+227} \lor \neg \left(y \leq 3 \cdot 10^{-61} \lor \neg \left(y \leq 5.2 \cdot 10^{-37}\right) \land y \leq 4 \cdot 10^{+58}\right):\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 8
Error30.9
Cost848
\[\begin{array}{l} t_1 := y \cdot \frac{z}{a}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \end{array} \]
Alternative 9
Error30.2
Cost848
\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -23500:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-148}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 10
Error0.7
Cost841
\[\begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{-73} \lor \neg \left(a \leq 10^{-8}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]
Alternative 11
Error10.2
Cost713
\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-16} \lor \neg \left(t \leq 4.6 \cdot 10^{-66}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \]
Alternative 12
Error10.2
Cost713
\[\begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-14} \lor \neg \left(t \leq 4.5 \cdot 10^{-66}\right):\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \]
Alternative 13
Error2.5
Cost576
\[x + \left(z - t\right) \cdot \frac{y}{a} \]
Alternative 14
Error30.9
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023031 
(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)))