Average Error: 0.0 → 0.0
Time: 5.5s
Precision: binary64
\[\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b \]
↓
\[a + \left(\mathsf{fma}\left(b, t + y, z + x\right) - \mathsf{fma}\left(z, y, \mathsf{fma}\left(a, t, b \cdot 2\right)\right)\right) \]
(FPCore (x y z t a b)
 :precision binary64
 (+ (- (- x (* (- y 1.0) z)) (* (- t 1.0) a)) (* (- (+ y t) 2.0) b)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (+ a (- (fma b (+ t y) (+ z x)) (fma z y (fma a t (* b 2.0))))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((x - ((y - 1.0) * z)) - ((t - 1.0) * a)) + (((y + t) - 2.0) * b);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return a + (fma(b, (t + y), (z + x)) - fma(z, y, fma(a, t, (b * 2.0))));
}

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x - Float64(Float64(y - 1.0) * z)) - Float64(Float64(t - 1.0) * a)) + Float64(Float64(Float64(y + t) - 2.0) * b))
end
↓
function code(x, y, z, t, a, b)
	return Float64(a + Float64(fma(b, Float64(t + y), Float64(z + x)) - fma(z, y, fma(a, t, Float64(b * 2.0)))))
end

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x - N[(N[(y - 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(N[(t - 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y + t), $MachinePrecision] - 2.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := N[(a + N[(N[(b * N[(t + y), $MachinePrecision] + N[(z + x), $MachinePrecision]), $MachinePrecision] - N[(z * y + N[(a * t + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b
↓
a + \left(\mathsf{fma}\left(b, t + y, z + x\right) - \mathsf{fma}\left(z, y, \mathsf{fma}\left(a, t, b \cdot 2\right)\right)\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.0

    \[\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + t\right) - 2, b, \mathsf{fma}\left(z, 1 - y, \mathsf{fma}\left(a, 1 - t, x\right)\right)\right)} \]

  3. Taylor expanded in y around 0 0.0

    \[\leadsto \color{blue}{\left(y \cdot b + \left(a + \left(z + \left(t \cdot b + x\right)\right)\right)\right) - \left(y \cdot z + \left(a \cdot t + 2 \cdot b\right)\right)} \]

  4. Simplified0.0

    \[\leadsto \color{blue}{a + \left(\mathsf{fma}\left(b, t + y, z + x\right) - \mathsf{fma}\left(z, y, \mathsf{fma}\left(a, t, b \cdot 2\right)\right)\right)} \]

  5. Final simplification0.0

    \[\leadsto a + \left(\mathsf{fma}\left(b, t + y, z + x\right) - \mathsf{fma}\left(z, y, \mathsf{fma}\left(a, t, b \cdot 2\right)\right)\right) \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z t a b)
  :name "Statistics.Distribution.Beta:$centropy from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (- x (* (- y 1.0) z)) (* (- t 1.0) a)) (* (- (+ y t) 2.0) b)))