Average Error: 6.5 → 1.7
Time: 9.3s
Precision: binary64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
↓
\[2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, b, a\right) \cdot \left(c \cdot \left(-i\right)\right)\right)\right) \]
(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))
↓
(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (fma z t (fma x y (* (fma c b a) (* c (- i)))))))
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * fma(z, t, fma(x, y, (fma(c, b, a) * (c * -i))));
}

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

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

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

Bits error versus c

Bits error versus i

Target

Original6.5
Target1.7
Herbie1.7
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \]

Derivation

  1. Initial program 6.5

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

  2. Simplified1.7

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

  3. Taylor expanded in b around 0 13.6

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

  4. Simplified6.5

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

  5. Applied egg-rr2.1

    \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, y \cdot x - \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)}\right)}^{3}}\right) \]

  6. Applied egg-rr1.7

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

  7. Final simplification1.7

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

Reproduce

herbie shell --seed 2022156 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (* 2.0 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i))))

  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))