?

Average Accuracy: 82.8% → 97.9%
Time: 14.5s
Precision: binary64
Cost: 6976.00

?

\[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
↓
\[\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right) \]
(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))
↓
(FPCore (x y z t a) :precision binary64 (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) {
	return fma(y, ((z - t) / (z - a)), x);
}

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)
	return fma(y, Float64(Float64(z - t) / Float64(z - a)), x)
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_] := N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)

Error?

Target

Original82.8%
Target98.1%
Herbie97.9%
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation?

  1. Initial program 82.8%

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

  2. Simplified97.9%

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

    [Start]82.8

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

    +-commutative [=>]82.8

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

    associate-*r/ [<=]97.9

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

    fma-def [=>]97.9

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

  3. Final simplification97.9%

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

Alternatives

Alternative 1
Accuracy62.5%
Cost1900.00
\[\begin{array}{l} t_1 := y \cdot \frac{-t}{z - a}\\ t_2 := x + \frac{y}{\frac{a}{t}}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-87}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-133}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-203}:\\ \;\;\;\;y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-295}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{-288}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{z}\\ \mathbf{elif}\;x \leq 7.3 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-213}:\\ \;\;\;\;y \cdot \frac{z - t}{z}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+94}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 2
Accuracy60.4%
Cost1900.00
\[\begin{array}{l} t_1 := y \cdot \frac{-t}{z - a}\\ \mathbf{if}\;x \leq -3 \cdot 10^{-87}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-134}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-204}:\\ \;\;\;\;y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;x \leq -1.38 \cdot 10^{-275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-295}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-288}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{z}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-256}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+93}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]
Alternative 3
Accuracy60.6%
Cost1900.00
\[\begin{array}{l} t_1 := y \cdot \frac{-t}{z - a}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-88}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-135}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-205}:\\ \;\;\;\;y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-295}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-288}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{z}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-256}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+93}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]
Alternative 4
Accuracy83.7%
Cost1168.00
\[\begin{array}{l} t_1 := x + \frac{y}{1 - \frac{a}{z}}\\ t_2 := x + \frac{z - t}{\frac{a}{-y}}\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-171}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Accuracy81.7%
Cost1105.00
\[\begin{array}{l} t_1 := x + \frac{y}{1 - \frac{a}{z}}\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-78}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+94} \lor \neg \left(x \leq 1.35 \cdot 10^{+135}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]
Alternative 6
Accuracy72.4%
Cost976.00
\[\begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+97}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-299}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-171}:\\ \;\;\;\;x + \frac{y}{\frac{-z}{t}}\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-92}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]
Alternative 7
Accuracy73.4%
Cost840.00
\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]
Alternative 8
Accuracy77.8%
Cost712.00
\[\begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+40}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+38}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 9
Accuracy77.8%
Cost712.00
\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+39}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+38}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 10
Accuracy95.5%
Cost704.00
\[x + \left(z - t\right) \cdot \frac{y}{z - a} \]
Alternative 11
Accuracy98.1%
Cost704.00
\[x + \frac{y}{\frac{z - a}{z - t}} \]
Alternative 12
Accuracy67.7%
Cost588.00
\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-277}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 13
Accuracy67.7%
Cost588.00
\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-211}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-277}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 14
Accuracy69.9%
Cost456.00
\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-23}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 15
Accuracy58.4%
Cost328.00
\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+177}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+157}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 16
Accuracy55.4%
Cost64.00
\[x \]

Error

Reproduce?

herbie shell --seed 2023096 
(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))))