Average Error: 1.4 → 1.4
Time: 6.5s
Precision: binary64
\[x + y \cdot \frac{z - t}{a - t} \]
↓
\[\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right) \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- a t)))))
↓
(FPCore (x y z t a) :precision binary64 (fma y (/ (- z t) (- a t)) x))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (a - t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	return fma(y, ((z - t) / (a - t)), x);
}

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.4
Target0.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  3. Final simplification1.4

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

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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