?

Average Error: 1.3 → 1.1
Time: 14.5s
Precision: binary64
Cost: 7240

?

\[x + y \cdot \frac{z - t}{z - a} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= y -9.2e+55)
   (+ x (* y (/ (- z t) (- z a))))
   (if (<= y 4.5e-24)
     (+ x (/ (* y (- z t)) (- z a)))
     (fma (- z t) (/ y (- z a)) x))))
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 <= -9.2e+55) {
		tmp = x + (y * ((z - t) / (z - a)));
	} else if (y <= 4.5e-24) {
		tmp = x + ((y * (z - t)) / (z - a));
	} else {
		tmp = fma((z - t), (y / (z - a)), x);
	}
	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 <= -9.2e+55)
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))));
	elseif (y <= 4.5e-24)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)));
	else
		tmp = fma(Float64(z - t), Float64(y / Float64(z - a)), x);
	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, -9.2e+55], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-24], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -9.2 \cdot 10^{+55}:\\
\;\;\;\;x + y \cdot \frac{z - t}{z - a}\\

\mathbf{elif}\;y \leq 4.5 \cdot 10^{-24}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\


\end{array}

Error?

Target

Original1.3
Target1.2
Herbie1.1
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation?

  1. Split input into 3 regimes

  2. if y < -9.1999999999999995e55

    1. Initial program 0.8

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

    if -9.1999999999999995e55 < y < 4.4999999999999997e-24

    1. Initial program 1.9

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

    2. Simplified0.6

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

      [Start]1.9

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

      associate-*r/ [=>]0.6

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

    if 4.4999999999999997e-24 < y

    1. Initial program 0.4

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

    2. Simplified2.5

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

      [Start]0.4

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

      +-commutative [=>]0.4

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

      associate-*r/ [=>]20.7

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

      *-commutative [=>]20.7

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

      associate-*r/ [<=]2.5

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

      fma-def [=>]2.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error10.7
Cost2640
\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -10000000:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-47}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;t_1 \leq 0.04:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\left(y + x\right) - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-y}{z - a}\\ \end{array} \]
Alternative 2
Error12.0
Cost2576
\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -10000000:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \mathbf{elif}\;t_1 \leq 1.00000000002:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;t_1 \leq 10^{+135}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+149}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-y}{z - a}\\ \end{array} \]
Alternative 3
Error12.0
Cost2576
\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -10000000:\\ \;\;\;\;\frac{t_2}{z - a}\\ \mathbf{elif}\;t_1 \leq 1.00000000002:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;t_1 \leq 10^{+135}:\\ \;\;\;\;x + \frac{t_2}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+149}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-y}{z - a}\\ \end{array} \]
Alternative 4
Error12.5
Cost2060
\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -10000000:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x - y \cdot \frac{z}{a}\\ \mathbf{elif}\;t_1 \leq 10^{+149}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-y}{z - a}\\ \end{array} \]
Alternative 5
Error22.4
Cost1768
\[\begin{array}{l} t_1 := \frac{y}{\frac{z - a}{z}}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-128}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-212}:\\ \;\;\;\;\frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-245}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-277}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-264}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-127}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-112}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z}\\ \mathbf{elif}\;x \leq 65:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 6
Error12.1
Cost1608
\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -10000000:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \mathbf{elif}\;t_1 \leq 10^{+149}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-y}{z - a}\\ \end{array} \]
Alternative 7
Error0.6
Cost969
\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+55} \lor \neg \left(y \leq 2 \cdot 10^{-56}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \end{array} \]
Alternative 8
Error14.3
Cost713
\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-29} \lor \neg \left(z \leq 1.4 \cdot 10^{-113}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array} \]
Alternative 9
Error14.5
Cost713
\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-30} \lor \neg \left(z \leq 2.3 \cdot 10^{-113}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]
Alternative 10
Error1.3
Cost704
\[x + y \cdot \frac{z - t}{z - a} \]
Alternative 11
Error19.8
Cost456
\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+163}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 12
Error26.9
Cost328
\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+142}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 13
Error28.5
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023039 
(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)))))