Average Error: 6.4 → 2.1
Time: 5.0s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot z}{t} \]
↓
\[\mathsf{fma}\left(y - x, \frac{z}{t}, x\right) \]
(FPCore (x y z t) :precision binary64 (+ x (/ (* (- y x) z) t)))
↓
(FPCore (x y z t) :precision binary64 (fma (- y x) (/ z t) x))
double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}
↓
double code(double x, double y, double z, double t) {
	return fma((y - x), (z / t), x);
}

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

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

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

Error

Target

Original6.4
Target2.3
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

  1. Initial program 6.4

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

  2. Simplified2.1

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

  3. Final simplification2.1

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

Reproduce

herbie shell --seed 2022210 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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