?

Average Error: 10.7 → 0.5
Time: 13.7s
Precision: binary64
Cost: 8264

?

\[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
↓
\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) (- z a))))
   (if (<= t_1 (- INFINITY))
     (+ x (/ y (/ (- z a) (- z t))))
     (if (<= t_1 5e+143)
       (- x (/ (* y (- t z)) (- z a)))
       (fma y (/ (- z t) (- 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 t_1 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + (y / ((z - a) / (z - t)));
	} else if (t_1 <= 5e+143) {
		tmp = x - ((y * (t - z)) / (z - a));
	} else {
		tmp = fma(y, ((z - t) / (z - a)), x);
	}
	return tmp;
}

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)))
end
↓
function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + Float64(y / Float64(Float64(z - a) / Float64(z - t))));
	elseif (t_1 <= 5e+143)
		tmp = Float64(x - Float64(Float64(y * Float64(t - z)) / Float64(z - a)));
	else
		tmp = fma(y, Float64(Float64(z - t) / Float64(z - a)), x);
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[(y / N[(N[(z - a), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+143], N[(x - N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\

\mathbf{elif}\;t_1 \leq 5 \cdot 10^{+143}:\\
\;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\

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


\end{array}

Error?

Target

Original10.7
Target1.2
Herbie0.5
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation?

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.1

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

      [Start]64.0

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

      associate-/l* [=>]0.1

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

    if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 5.00000000000000012e143

    1. Initial program 0.2

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

    if 5.00000000000000012e143 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

    1. Initial program 39.1

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

    2. Simplified2.6

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

      [Start]39.1

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

      +-commutative [=>]39.1

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

      associate-*r/ [<=]2.6

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

      fma-def [=>]2.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.4
Cost1993
\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 4 \cdot 10^{+221}\right):\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\ \end{array} \]
Alternative 2
Error23.0
Cost1900
\[\begin{array}{l} t_1 := x + t \cdot \frac{y}{a}\\ t_2 := \frac{y \cdot z}{z - a}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-128}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-308}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-263}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{z - a}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{-233}:\\ \;\;\;\;x - y \cdot \frac{t}{z}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-216}:\\ \;\;\;\;z \cdot \frac{y}{z - a}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-127}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z}\\ \mathbf{elif}\;x \leq 13:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 3
Error14.8
Cost1104
\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-29}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-114}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 9000:\\ \;\;\;\;x - y \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+25}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 4
Error12.6
Cost1104
\[\begin{array}{l} t_1 := x + y \cdot \left(1 - \frac{t}{z}\right)\\ t_2 := x + y \cdot \frac{t - z}{a}\\ \mathbf{if}\;a \leq -56000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-24}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error1.5
Cost969
\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-8} \lor \neg \left(z \leq 6.4 \cdot 10^{-255}\right):\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array} \]
Alternative 6
Error21.0
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-259}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-253}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{+163}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 7
Error12.1
Cost841
\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+18} \lor \neg \left(z \leq 6 \cdot 10^{-134}\right):\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]
Alternative 8
Error3.4
Cost836
\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+108}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array} \]
Alternative 9
Error14.4
Cost713
\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-29} \lor \neg \left(z \leq 2.6 \cdot 10^{-113}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]
Alternative 10
Error14.5
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-29}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-113}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 11
Error19.8
Cost456
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+163}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 12
Error26.9
Cost328
\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+145}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
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:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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