Average Error: 0.1 → 0.1
Time: 32.2s
Precision: binary64
Cost: 32832
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
↓
\[\mathsf{fma}\left(y, i, \mathsf{fma}\left(b + -0.5, \log c, \mathsf{fma}\left(\log y, x, z + t\right) + a\right)\right) \]
(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))
↓
(FPCore (x y z t a b c i)
 :precision binary64
 (fma y i (fma (+ b -0.5) (log c) (+ (fma (log y) x (+ z t)) a))))
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(y, i, fma((b + -0.5), log(c), (fma(log(y), x, (z + t)) + a)));
}

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end
↓
function code(x, y, z, t, a, b, c, i)
	return fma(y, i, fma(Float64(b + -0.5), log(c), Float64(fma(log(y), x, Float64(z + t)) + a)))
end

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i + N[(N[(b + -0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x + N[(z + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i
↓
\mathsf{fma}\left(y, i, \mathsf{fma}\left(b + -0.5, \log c, \mathsf{fma}\left(\log y, x, z + t\right) + a\right)\right)

Error

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(b + -0.5, \log c, \mathsf{fma}\left(\log y, x, z + t\right) + a\right)\right)} \]
    Proof

Alternatives

Alternative 1
Error6.7
Cost14156
\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ t_2 := t_1 + \left(a + \left(\log y \cdot x + \left(t + z\right)\right)\right)\\ \mathbf{if}\;y \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \left(\log c \cdot \left(b - 0.5\right) + \left(t + z\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(a + \left(t + z\right)\right)\right) + y \cdot i\\ \end{array} \]
Alternative 2
Error0.5
Cost14152
\[\begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \log c \cdot b\right) + y \cdot i\\ \mathbf{if}\;x \leq -2 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \left(\log c \cdot \left(b - 0.5\right) + \left(t + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error0.1
Cost14016
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
Alternative 4
Error5.2
Cost13896
\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ t_2 := t_1 + \left(a + \left(\log y \cdot x + t\right)\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+211}:\\ \;\;\;\;\left(t_1 + \left(a + \left(t + z\right)\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error5.2
Cost13896
\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c + \left(a + \left(\log y \cdot x + t\right)\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \left(t + z\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error5.2
Cost13896
\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c + \left(a + \left(\log y \cdot x + t\right)\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \left(\log c \cdot \left(b - 0.5\right) + \left(t + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error6.9
Cost13636
\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -5 \cdot 10^{+206}:\\ \;\;\;\;t_1 + \left(t + \left(z + -0.5 \cdot \log c\right)\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(t + z\right)\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_1 + y \cdot i\\ \end{array} \]
Alternative 8
Error7.0
Cost13636
\[\begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+170}:\\ \;\;\;\;t_2 + \left(t_1 + t\right)\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+228}:\\ \;\;\;\;\left(t_2 + \left(a + \left(t + z\right)\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_1 + y \cdot i\\ \end{array} \]
Alternative 9
Error26.9
Cost8164
\[\begin{array}{l} t_1 := a + \left(t + \left(z + -0.5 \cdot \log c\right)\right)\\ t_2 := \log c \cdot b + y \cdot i\\ t_3 := \log y \cdot x + y \cdot i\\ \mathbf{if}\;b \leq -3 \cdot 10^{+194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -7.3 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4 \cdot 10^{+130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.45 \cdot 10^{+60}:\\ \;\;\;\;a + y \cdot i\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{+52}:\\ \;\;\;\;t + y \cdot i\\ \mathbf{elif}\;b \leq -3.3 \cdot 10^{-15}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-233}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 10
Error38.3
Cost8036
\[\begin{array}{l} t_1 := t + y \cdot i\\ t_2 := \log c \cdot b + y \cdot i\\ t_3 := a + y \cdot i\\ t_4 := z + y \cdot i\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+168}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-168}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-229}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.48 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 520000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+209}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]
Alternative 11
Error38.5
Cost8036
\[\begin{array}{l} t_1 := \log c \cdot b + y \cdot i\\ t_2 := a + y \cdot i\\ t_3 := z + y \cdot i\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+169}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-242}:\\ \;\;\;\;\log y \cdot x + y \cdot i\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-229}:\\ \;\;\;\;a\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-98}:\\ \;\;\;\;t + y \cdot i\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+209}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 12
Error7.4
Cost7624
\[\begin{array}{l} t_1 := \log y \cdot x + y \cdot i\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(t + z\right)\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 13
Error14.1
Cost7496
\[\begin{array}{l} t_1 := a + \left(t + z\right)\\ \mathbf{if}\;a \leq -9.2 \cdot 10^{+101}:\\ \;\;\;\;\log c \cdot b + t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\left(\log c \cdot \left(b - 0.5\right) + \left(t + z\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + t_1\\ \end{array} \]
Alternative 14
Error14.9
Cost7368
\[\begin{array}{l} t_1 := \log y \cdot x + y \cdot i\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{+191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+228}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(a + \left(t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 15
Error39.1
Cost7252
\[\begin{array}{l} t_1 := a + y \cdot i\\ t_2 := z + y \cdot i\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-289}:\\ \;\;\;\;t + y \cdot i\\ \mathbf{elif}\;z \leq 15500000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+87}:\\ \;\;\;\;\log c \cdot b\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+209}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 16
Error15.9
Cost7240
\[\begin{array}{l} t_1 := \log y \cdot x + y \cdot i\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+228}:\\ \;\;\;\;\log c \cdot b + \left(a + \left(t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 17
Error46.2
Cost1116
\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+87}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq -1.12 \cdot 10^{-125}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-260}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-300}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-101}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 4500000:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+26}:\\ \;\;\;\;y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
Alternative 18
Error39.1
Cost716
\[\begin{array}{l} t_1 := a + y \cdot i\\ \mathbf{if}\;a \leq -3 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-123}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 4800000:\\ \;\;\;\;t + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 19
Error37.6
Cost716
\[\begin{array}{l} t_1 := a + y \cdot i\\ \mathbf{if}\;a \leq -2 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-171}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;a \leq 4800000:\\ \;\;\;\;t + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 20
Error39.2
Cost584
\[\begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+239}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+209}:\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
Alternative 21
Error44.6
Cost460
\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+88}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 3.15 \cdot 10^{+94}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
Alternative 22
Error44.5
Cost328
\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+88}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+94}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
Alternative 23
Error51.8
Cost64
\[a \]

Error

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))