?

Average Accuracy: 84.1% → 97.7%
Time: 13.9s
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

Original84.1%
Target97.9%
Herbie97.7%
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation?

  1. Initial program 84.1%

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

  2. Simplified97.7%

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

    [Start]84.1

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

    +-commutative [=>]84.1

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

    associate-*r/ [<=]97.7

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

    fma-def [=>]97.7

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

  3. Final simplification97.7%

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

Alternatives

Alternative 1
Accuracy84.0%
Cost1501
\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z}}\\ t_2 := x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-105}:\\ \;\;\;\;x - \frac{y \cdot t}{z - a}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-272}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-133}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+49} \lor \neg \left(z \leq 2 \cdot 10^{+164}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 2
Accuracy84.3%
Cost1369
\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z}}\\ t_2 := x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.7 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-104}:\\ \;\;\;\;x - \frac{y \cdot t}{z - a}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-133}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+49} \lor \neg \left(z \leq 2 \cdot 10^{+163}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 3
Accuracy67.9%
Cost1108
\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-48}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+34}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+80}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 4
Accuracy84.0%
Cost1104
\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{if}\;z \leq -3.55 \cdot 10^{+93}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;x - y \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Accuracy67.4%
Cost1044
\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-51}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+80}:\\ \;\;\;\;t \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 6
Accuracy67.4%
Cost1044
\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-47}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+34}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+80}:\\ \;\;\;\;\frac{-t}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 7
Accuracy77.2%
Cost976
\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-93}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+65}:\\ \;\;\;\;x - y \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 8
Accuracy76.5%
Cost976
\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-93}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-36}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;x - y \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 9
Accuracy81.3%
Cost841
\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-104} \lor \neg \left(z \leq 8.5 \cdot 10^{-38}\right):\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]
Alternative 10
Accuracy77.0%
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-93}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-21}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 11
Accuracy77.2%
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-93}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 12
Accuracy95.0%
Cost704
\[x - \frac{y}{z - a} \cdot \left(t - z\right) \]
Alternative 13
Accuracy97.9%
Cost704
\[x + \frac{y}{\frac{z - a}{z - t}} \]
Alternative 14
Accuracy68.9%
Cost456
\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-47}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 15
Accuracy57.1%
Cost328
\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-279}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-192}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 16
Accuracy54.6%
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023133 
(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))))