?

Average Error: 2.07% → 1.98%
Time: 10.9s
Precision: binary64
Cost: 7108

?

\[x + y \cdot \frac{z - t}{z - a} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-217}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= y -2.7e+46)
   (fma (- z t) (/ y (- z a)) x)
   (if (<= y 7.5e-217)
     (+ x (/ (* y (- z t)) (- z a)))
     (+ x (* y (/ (- z t) (- z a)))))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.7e+46) {
		tmp = fma((z - t), (y / (z - a)), x);
	} else if (y <= 7.5e-217) {
		tmp = x + ((y * (z - t)) / (z - a));
	} else {
		tmp = x + (y * ((z - t) / (z - a)));
	}
	return tmp;
}

function code(x, y, z, t, a)
	return Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))))
end
↓
function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2.7e+46)
		tmp = fma(Float64(z - t), Float64(y / Float64(z - a)), x);
	elseif (y <= 7.5e-217)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))));
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2.7e+46], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 7.5e-217], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -2.7 \cdot 10^{+46}:\\
\;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\

\mathbf{elif}\;y \leq 7.5 \cdot 10^{-217}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\

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


\end{array}

Error?

Target

Original2.07%
Target1.82%
Herbie1.98%
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation?

  1. Split input into 3 regimes

  2. if y < -2.7000000000000002e46

    1. Initial program 1.15

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

    2. Simplified4.9

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

      [Start]1.15

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

      +-commutative [=>]1.15

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

      associate-*r/ [=>]42.6

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

      *-commutative [=>]42.6

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

      associate-*r/ [<=]4.9

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

      fma-def [=>]4.9

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

    if -2.7000000000000002e46 < y < 7.50000000000000031e-217

    1. Initial program 3.15

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

    2. Simplified1.09

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

      [Start]3.15

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

      associate-*r/ [=>]1.09

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

    if 7.50000000000000031e-217 < y

    1. Initial program 1.48

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

  4. Final simplification1.98

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-217}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternatives

Alternative 1
Error22.25%
Cost1236
\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z}\\ \mathbf{if}\;t \leq -8.4 \cdot 10^{+247}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{+173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{+51}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+28}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;t \leq 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]
Alternative 2
Error1.26%
Cost969
\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+46} \lor \neg \left(y \leq 6.5 \cdot 10^{-217}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \end{array} \]
Alternative 3
Error33.53%
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{+73}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+202}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 4
Error16.1%
Cost841
\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-87} \lor \neg \left(z \leq 3.2 \cdot 10^{-161}\right):\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]
Alternative 5
Error12.44%
Cost841
\[\begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+50} \lor \neg \left(t \leq 1.6 \cdot 10^{-13}\right):\\ \;\;\;\;x - y \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \end{array} \]
Alternative 6
Error21.59%
Cost713
\[\begin{array}{l} \mathbf{if}\;z \leq -3800 \lor \neg \left(z \leq 2.7 \cdot 10^{+18}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array} \]
Alternative 7
Error21.55%
Cost713
\[\begin{array}{l} \mathbf{if}\;z \leq -7800000 \lor \neg \left(z \leq 1.38 \cdot 10^{+17}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]
Alternative 8
Error2.07%
Cost704
\[x + y \cdot \frac{z - t}{z - a} \]
Alternative 9
Error30.05%
Cost456
\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 10
Error41.88%
Cost328
\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-256}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-110}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 11
Error44%
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023121 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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