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

Average Accuracy: 99.8% → 99.9%

Time: 25.1s

Precision: binary64

Cost: 13632

Math

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

↓

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (fma (+ a -0.5) b (+ z (- (+ x y) (* z (log t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	return fma((a + -0.5), b, (z + ((x + y) - (z * log(t)))));
}

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	99.8%
Target	99.4%
Herbie	99.9%

\[\left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \]

Derivation

1. Initial program 99.8%

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

2. Simplified 99.9%

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

   [Start] 99.8	\[ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   +-commutative [=>] 99.8	\[ \color{blue}{\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(a - 0.5, b, \left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]
   sub-neg [=>] 99.9	\[ \mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, b, \left(\left(x + y\right) + z\right) - z \cdot \log t\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(a + \color{blue}{-0.5}, b, \left(\left(x + y\right) + z\right) - z \cdot \log t\right) \]
   +-commutative [=>] 99.9	\[ \mathsf{fma}\left(a + -0.5, b, \color{blue}{\left(z + \left(x + y\right)\right)} - z \cdot \log t\right) \]
   associate--l+ [=>] 99.9	\[ \mathsf{fma}\left(a + -0.5, b, \color{blue}{z + \left(\left(x + y\right) - z \cdot \log t\right)}\right) \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	65.9%
Cost	9564

\[\begin{array}{l} t_1 := -0.5 \cdot b + \left(y + \left(z + x\right)\right)\\ t_2 := \left(z + y\right) - z \cdot \log t\\ t_3 := \left(a + -0.5\right) \cdot b\\ t_4 := x + z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, x\right)\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-124}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-228}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-273}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;y + t_3\\ \end{array} \]

Alternative 2
Accuracy	89.9%
Cost	8008

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+223}:\\ \;\;\;\;t_1 + z \cdot \left(1 - \log t\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+203}:\\ \;\;\;\;\left(-0.5 \cdot b + \left(y + \left(z + x\right)\right)\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 3
Accuracy	84.0%
Cost	7752

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \mathbf{elif}\;t_1 \leq 10^{+100}:\\ \;\;\;\;\left(x + y\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 4
Accuracy	84.0%
Cost	7752

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \mathbf{elif}\;t_1 \leq 10^{+100}:\\ \;\;\;\;\left(y + \left(z + x\right)\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 5
Accuracy	71.5%
Cost	7640

\[\begin{array}{l} t_1 := -0.5 \cdot b + \left(y + \left(z + x\right)\right)\\ t_2 := x + z \cdot \left(1 - \log t\right)\\ t_3 := y + \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-277}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+50}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	64.6%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;x + y \leq -4 \cdot 10^{-41}:\\ \;\;\;\;\left(-0.5 \cdot b + \left(z + x\right)\right) - z \cdot \log t\\ \mathbf{elif}\;x + y \leq 10^{+46}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \end{array} \]

Alternative 7
Accuracy	69.3%
Cost	7512

\[\begin{array}{l} t_1 := -0.5 \cdot b + \left(y + \left(z + x\right)\right)\\ t_2 := z \cdot \left(1 - \log t\right)\\ t_3 := y + \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;z \leq -2.45 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-277}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+52}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	80.1%
Cost	7500

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ t_2 := y + t_1\\ \mathbf{if}\;b \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+53}:\\ \;\;\;\;\left(y + \left(z + x\right)\right) - z \cdot \log t\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{+176}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 + z \cdot \left(1 - \log t\right)\\ \end{array} \]

Alternative 9
Accuracy	99.8%
Cost	7488

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

Alternative 10
Accuracy	99.8%
Cost	7360

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

Alternative 11
Accuracy	54.4%
Cost	1360

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x + y \leq 4 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq 4 \cdot 10^{+177}:\\ \;\;\;\;y\\ \mathbf{elif}\;x + y \leq 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12
Accuracy	47.5%
Cost	1360

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{+17}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 4 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq 4 \cdot 10^{+177}:\\ \;\;\;\;y\\ \mathbf{elif}\;x + y \leq 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 13
Accuracy	46.8%
Cost	1360

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -4 \cdot 10^{-41}:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 4 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq 4 \cdot 10^{+177}:\\ \;\;\;\;y\\ \mathbf{elif}\;x + y \leq 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 14
Accuracy	53.1%
Cost	1100

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq 4 \cdot 10^{+48}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;x + y \leq 4 \cdot 10^{+177}:\\ \;\;\;\;y\\ \mathbf{elif}\;x + y \leq 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 15
Accuracy	52.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x + y \leq 4 \cdot 10^{-85}:\\ \;\;\;\;a \cdot b + \left(x + -0.5 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 16
Accuracy	52.1%
Cost	720

\[\begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+231}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;b \leq -2.9 \cdot 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+181}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \]

Alternative 17
Accuracy	52.7%
Cost	708

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq 4 \cdot 10^{-85}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 18
Accuracy	29.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-250}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+40}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 19
Accuracy	31.7%
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 20
Accuracy	24.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))