?

Average Accuracy: 85.9% → 98.2%
Time: 15.0s
Precision: binary64
Cost: 6976

?

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

Original85.9%
Target98.4%
Herbie98.2%
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation?

  1. Initial program 83.9%

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

  2. Simplified98.8%

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

    [Start]83.9

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

    +-commutative [=>]83.9

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

    associate-*r/ [<=]98.7

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

    fma-def [=>]98.8

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

  3. Final simplification98.8%

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

Alternatives

Alternative 1
Accuracy70.0%
Cost2029
\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ t_2 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+103}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-67}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-215}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-247}:\\ \;\;\;\;x - \frac{y \cdot t}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+14} \lor \neg \left(y \leq 2.6 \cdot 10^{+90}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 2
Accuracy73.3%
Cost1766
\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+98}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;y \leq -8.4 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-67}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-108} \lor \neg \left(y \leq 1.35 \cdot 10^{-15} \lor \neg \left(y \leq 105000\right) \land y \leq 1.6 \cdot 10^{+93}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{z}}\\ \end{array} \]
Alternative 3
Accuracy59.8%
Cost1244
\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+68}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+28}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-80}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-297}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 10^{-268}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 4
Accuracy60.6%
Cost980
\[\begin{array}{l} t_1 := y \cdot \frac{t}{a}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{-83}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-298}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 5
Accuracy60.6%
Cost980
\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-84}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-297}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-268}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 6
Accuracy60.7%
Cost980
\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-80}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-297}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-271}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 7
Accuracy74.8%
Cost976
\[\begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+68}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+28}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-58}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+86}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 8
Accuracy74.8%
Cost976
\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+68}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -7 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-57}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+90}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 9
Accuracy79.3%
Cost972
\[\begin{array}{l} t_1 := x + \frac{y}{1 - \frac{a}{z}}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-123}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+86}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Accuracy75.3%
Cost844
\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+100}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-123}:\\ \;\;\;\;x - \frac{y \cdot t}{z}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+88}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 11
Accuracy95.8%
Cost704
\[x + \left(z - t\right) \cdot \frac{y}{z - a} \]
Alternative 12
Accuracy98.4%
Cost704
\[x + \frac{y}{\frac{z - a}{z - t}} \]
Alternative 13
Accuracy62.7%
Cost456
\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-81}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 14
Accuracy50.1%
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023159 
(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))))