Average Error: 2.7 → 2.7
Time: 5.9s
Precision: binary64
\[\frac{x}{y - z \cdot t} \]
↓
\[\frac{x}{\mathsf{fma}\left(z, -t, y\right)} \]
(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))
↓
(FPCore (x y z t) :precision binary64 (/ x (fma z (- t) y)))
double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}
↓
double code(double x, double y, double z, double t) {
	return x / fma(z, -t, y);
}

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end
↓
function code(x, y, z, t)
	return Float64(x / fma(z, Float64(-t), y))
end

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.7
Target1.7
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array} \]

Derivation

  1. Initial program 2.7

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

  2. Applied egg-rr2.7

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

  3. Final simplification2.7

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

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

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