Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B

Average Error: 0.1 → 0.1

Time: 24.0s

Precision: binary64

Cost: 32832

\[ \begin{array}{c}[z, t, a] = \mathsf{sort}([z, t, a])\\ \end{array} \]

Math

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

↓

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

FPCore

(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
 (+ z (fma (+ b -0.5) (log c) (fma y i (fma x (log y) (+ t a))))))

C

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 z + fma((b + -0.5), log(c), fma(y, i, fma(x, log(y), (t + a))));
}

Julia

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 Float64(z + fma(Float64(b + -0.5), log(c), fma(y, i, fma(x, log(y), Float64(t + a)))))
end

Wolfram

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[(z + N[(N[(b + -0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + N[(x * N[Log[y], $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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. Simplified 0.1

   \[\leadsto \color{blue}{z + \mathsf{fma}\left(b + -0.5, \log c, \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, t + a\right)\right)\right)} \]
   Proof
   (+.f64 z (fma.f64 (+.f64 b -1/2) (log.f64 c) (fma.f64 y i (fma.f64 x (log.f64 y) (+.f64 t a))))): 0 points increase in error, 0 points decrease in error
   (+.f64 z (fma.f64 (+.f64 b (Rewrite<= metadata-eval (neg.f64 1/2))) (log.f64 c) (fma.f64 y i (fma.f64 x (log.f64 y) (+.f64 t a))))): 0 points increase in error, 0 points decrease in error
   (+.f64 z (fma.f64 (Rewrite<= sub-neg_binary64 (-.f64 b 1/2)) (log.f64 c) (fma.f64 y i (fma.f64 x (log.f64 y) (+.f64 t a))))): 0 points increase in error, 0 points decrease in error
   (+.f64 z (fma.f64 (-.f64 b 1/2) (log.f64 c) (fma.f64 y i (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (log.f64 y)) (+.f64 t a)))))): 3 points increase in error, 0 points decrease in error
   (+.f64 z (fma.f64 (-.f64 b 1/2) (log.f64 c) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y i) (+.f64 (*.f64 x (log.f64 y)) (+.f64 t a)))))): 2 points increase in error, 0 points decrease in error
   (+.f64 z (fma.f64 (-.f64 b 1/2) (log.f64 c) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) (+.f64 t a)) (*.f64 y i))))): 0 points increase in error, 0 points decrease in error
   (+.f64 z (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 b 1/2) (log.f64 c)) (+.f64 (+.f64 (*.f64 x (log.f64 y)) (+.f64 t a)) (*.f64 y i))))): 2 points increase in error, 1 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 z (*.f64 (-.f64 b 1/2) (log.f64 c))) (+.f64 (+.f64 (*.f64 x (log.f64 y)) (+.f64 t a)) (*.f64 y i)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (-.f64 b 1/2) (log.f64 c)) z)) (+.f64 (+.f64 (*.f64 x (log.f64 y)) (+.f64 t a)) (*.f64 y i))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 (-.f64 b 1/2) (log.f64 c)) z) (+.f64 (*.f64 x (log.f64 y)) (+.f64 t a))) (*.f64 y i))): 1 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (-.f64 b 1/2) (log.f64 c)) (+.f64 z (+.f64 (*.f64 x (log.f64 y)) (+.f64 t a))))) (*.f64 y i)): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (-.f64 b 1/2) (log.f64 c)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 z (*.f64 x (log.f64 y))) (+.f64 t a)))) (*.f64 y i)): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (-.f64 b 1/2) (log.f64 c)) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x (log.f64 y)) z)) (+.f64 t a))) (*.f64 y i)): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (-.f64 b 1/2) (log.f64 c)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a))) (*.f64 y i)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b 1/2) (log.f64 c)))) (*.f64 y i)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	32832

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

Alternative 2
Error	2.8
Cost	14024

\[\begin{array}{l} t_1 := \log c \cdot \left(b + -0.5\right)\\ t_2 := x \cdot \log y\\ \mathbf{if}\;x \leq -9 \cdot 10^{+142}:\\ \;\;\;\;t_2 + \left(a + \left(z + t_1\right)\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+69}:\\ \;\;\;\;z + \left(y \cdot i + \left(a + \left(t + t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(a + \left(t_2 + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	0.1
Cost	14016

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

Alternative 4
Error	2.9
Cost	13896

\[\begin{array}{l} t_1 := \log c \cdot \left(b + -0.5\right)\\ t_2 := x \cdot \log y + \left(a + \left(z + t_1\right)\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+69}:\\ \;\;\;\;z + \left(y \cdot i + \left(a + \left(t + t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	5.2
Cost	13384

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, z + a\right)\\ \mathbf{if}\;x \leq -1.22 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+113}:\\ \;\;\;\;z + \left(y \cdot i + \left(a + \left(t + \log c \cdot \left(b + -0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	16.5
Cost	8016

\[\begin{array}{l} t_1 := a + \log c \cdot \left(b + -0.5\right)\\ \mathbf{if}\;b + -0.5 \leq -1 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b + -0.5 \leq 2 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \log y + \left(z + a\right)\\ \mathbf{elif}\;b + -0.5 \leq 5 \cdot 10^{+177}:\\ \;\;\;\;z + b \cdot \log c\\ \mathbf{elif}\;b + -0.5 \leq 5 \cdot 10^{+200}:\\ \;\;\;\;y \cdot i + \left(a + -0.5 \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.4
Cost	7888

\[\begin{array}{l} t_1 := x \cdot \log y + \left(z + a\right)\\ t_2 := a + \log c \cdot \left(b + -0.5\right)\\ \mathbf{if}\;b + -0.5 \leq -1 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b + -0.5 \leq 2 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b + -0.5 \leq 5 \cdot 10^{+172}:\\ \;\;\;\;z + b \cdot \log c\\ \mathbf{elif}\;b + -0.5 \leq 2 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	27.7
Cost	7776

\[\begin{array}{l} t_1 := a + \log c \cdot \left(b + -0.5\right)\\ t_2 := x \cdot \log y\\ t_3 := z + \left(t + a\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+188}:\\ \;\;\;\;z + t_2\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-298}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-178}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-140}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;z + b \cdot \log c\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+136}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;a + t_2\\ \end{array} \]

Alternative 9
Error	15.3
Cost	7764

\[\begin{array}{l} t_1 := z + b \cdot \log c\\ t_2 := y \cdot i + \left(a + \log c \cdot \left(b + -0.5\right)\right)\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{+95}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \log y + \left(z + a\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+224}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	5.2
Cost	7624

\[\begin{array}{l} t_1 := x \cdot \log y + \left(z + a\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;z + \left(y \cdot i + \left(a + \left(t + \log c \cdot \left(b + -0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	27.5
Cost	7384

\[\begin{array}{l} t_1 := z + \left(t + a\right)\\ t_2 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-140}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+229}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	26.7
Cost	7380

\[\begin{array}{l} t_1 := z + b \cdot \log c\\ t_2 := a + x \cdot \log y\\ t_3 := z + \left(t + a\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+167}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-273}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+136}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	25.4
Cost	7248

\[\begin{array}{l} t_1 := z + \left(t + a\right)\\ t_2 := a + x \cdot \log y\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+167}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-140}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	26.5
Cost	7248

\[\begin{array}{l} t_1 := z + \left(t + a\right)\\ t_2 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+188}:\\ \;\;\;\;z + t_2\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.0007:\\ \;\;\;\;z + b \cdot \log c\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a + t_2\\ \end{array} \]

Alternative 15
Error	26.9
Cost	6856

\[\begin{array}{l} t_1 := b \cdot \log c\\ \mathbf{if}\;b \leq -5.3 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+221}:\\ \;\;\;\;z + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	32.6
Cost	720

\[\begin{array}{l} \mathbf{if}\;i \leq -1.9 \cdot 10^{+155}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;i \leq -1.75 \cdot 10^{+118}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-27}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;i \leq 7.5:\\ \;\;\;\;y \cdot i\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \]

Alternative 17
Error	27.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+61}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \]

Alternative 18
Error	27.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 1.12 \cdot 10^{+61}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + a\right)\\ \end{array} \]

Alternative 19
Error	36.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+88}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \]

Alternative 20
Error	36.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+88}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 21
Error	47.2
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022330 
(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)))