Average Error: 0.0 → 0.0
Time: 12.0s
Precision: binary64
Cost: 19776
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
↓
\[\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right) \]
(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))
↓
(FPCore (x y z t a b c i)
 :precision binary64
 (fma x y (fma z t (fma c i (* a b)))))
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(x, y, fma(z, t, fma(c, i, (a * b))));
}

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

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

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

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)} \]
    Proof
    (fma.f64 x y (fma.f64 z t (fma.f64 c i (*.f64 a b)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 x y (fma.f64 z t (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c i) (*.f64 a b))))): 1 points increase in error, 0 points decrease in error
    (fma.f64 x y (fma.f64 z t (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 a b) (*.f64 c i))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 x y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z t) (+.f64 (*.f64 a b) (*.f64 c i))))): 1 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 z t) (+.f64 (*.f64 a b) (*.f64 c i))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (+.f64 (*.f64 a b) (*.f64 c i)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error26.8
Cost2008
\[\begin{array}{l} t_1 := c \cdot i + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -2.70639484167812 \cdot 10^{+55}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -3.216921093739587 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-320}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.2963300845675692 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 1.1477818983799606 \cdot 10^{-96}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 3.684930364342622 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
Alternative 2
Error24.2
Cost2008
\[\begin{array}{l} t_1 := c \cdot i + x \cdot y\\ t_2 := a \cdot b + c \cdot i\\ \mathbf{if}\;a \cdot b \leq -8.964551718449584 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq -3.216921093739587 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-320}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.2963300845675692 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 1.1477818983799606 \cdot 10^{-96}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 3.684930364342622 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 3
Error22.5
Cost2008
\[\begin{array}{l} t_1 := c \cdot i + z \cdot t\\ t_2 := c \cdot i + x \cdot y\\ t_3 := a \cdot b + c \cdot i\\ \mathbf{if}\;a \cdot b \leq -2.70639484167812 \cdot 10^{+55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-320}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 1.1477818983799606 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 3.684930364342622 \cdot 10^{+76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 2.4995343008912378 \cdot 10^{+104}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 4
Error22.1
Cost2008
\[\begin{array}{l} t_1 := c \cdot i + z \cdot t\\ t_2 := c \cdot i + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -2.70639484167812 \cdot 10^{+55}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-320}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 1.1477818983799606 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 219660442780658.6:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 7.016688697307986 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
Alternative 5
Error21.9
Cost2008
\[\begin{array}{l} t_1 := a \cdot b + z \cdot t\\ t_2 := a \cdot b + x \cdot y\\ \mathbf{if}\;c \cdot i \leq -6.552475320183716 \cdot 10^{+67}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -9.235571947182081 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq -1.1964602259079888 \cdot 10^{-224}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 1.788175022096186 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 2.702253921001196 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 2.955115648612862 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \end{array} \]
Alternative 6
Error37.3
Cost1752
\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -6.552475320183716 \cdot 10^{+67}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -9.235571947182081 \cdot 10^{-87}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -4.063910719074392 \cdot 10^{-305}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.228446676082557 \cdot 10^{-258}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 9.067602372408774 \cdot 10^{-213}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 2.955115648612862 \cdot 10^{+56}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
Alternative 7
Error37.8
Cost1492
\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.70639484167812 \cdot 10^{+55}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 2.9352658655117442 \cdot 10^{-276}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.2963300845675692 \cdot 10^{-159}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 1.1477818983799606 \cdot 10^{-96}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 7.016688697307986 \cdot 10^{+45}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
Alternative 8
Error17.7
Cost1232
\[\begin{array}{l} t_1 := c \cdot i + \left(a \cdot b + z \cdot t\right)\\ t_2 := a \cdot b + x \cdot y\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+234}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.81942464379364 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \end{array} \]
Alternative 9
Error6.6
Cost1224
\[\begin{array}{l} t_1 := c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{if}\;c \cdot i \leq -497.7125978490741:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 2.855395496097922 \cdot 10^{+75}:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error6.1
Cost1224
\[\begin{array}{l} t_1 := c \cdot i + \left(a \cdot b + x \cdot y\right)\\ \mathbf{if}\;c \cdot i \leq -6.552475320183716 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 9.794129869804682 \cdot 10^{+65}:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error0.0
Cost960
\[c \cdot i + \left(a \cdot b + \left(x \cdot y + z \cdot t\right)\right) \]
Alternative 12
Error37.7
Cost712
\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -6.552475320183716 \cdot 10^{+67}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 9.794129869804682 \cdot 10^{+65}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
Alternative 13
Error47.1
Cost192
\[c \cdot i \]

Error

Reproduce

herbie shell --seed 2022295 
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))