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

Average Accuracy: 99.8% → 99.9%

Time: 21.7s

Precision: binary64

Cost: 19904

Math

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

↓

\[x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]

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
 (+ x (fma z (- 1.0 (log t)) (fma (+ a -0.5) b y))))

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 x + fma(z, (1.0 - log(t)), fma((a + -0.5), b, y));
}

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 Float64(x + fma(z, Float64(1.0 - log(t)), fma(Float64(a + -0.5), b, y)))
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[(x + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b + y), $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%

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, y\right)\right)} \]
   Proof

   [Start] 99.8	\[ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   sub-neg [=>] 99.8	\[ \color{blue}{\left(\left(\left(x + y\right) + z\right) + \left(-z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 99.8	\[ \left(\color{blue}{\left(x + \left(y + z\right)\right)} + \left(-z \cdot \log t\right)\right) + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 99.8	\[ \color{blue}{\left(x + \left(\left(y + z\right) + \left(-z \cdot \log t\right)\right)\right)} + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 99.8	\[ \color{blue}{x + \left(\left(\left(y + z\right) + \left(-z \cdot \log t\right)\right) + \left(a - 0.5\right) \cdot b\right)} \]
   sub-neg [<=] 99.8	\[ x + \left(\color{blue}{\left(\left(y + z\right) - z \cdot \log t\right)} + \left(a - 0.5\right) \cdot b\right) \]
   associate-+r- [<=] 99.8	\[ x + \left(\color{blue}{\left(y + \left(z - z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b\right) \]
   +-commutative [=>] 99.8	\[ x + \left(\color{blue}{\left(\left(z - z \cdot \log t\right) + y\right)} + \left(a - 0.5\right) \cdot b\right) \]
   associate-+l+ [=>] 99.8	\[ x + \color{blue}{\left(\left(z - z \cdot \log t\right) + \left(y + \left(a - 0.5\right) \cdot b\right)\right)} \]
   +-commutative [<=] 99.8	\[ x + \left(\left(z - z \cdot \log t\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot b + y\right)}\right) \]
   sub-neg [=>] 99.8	\[ x + \left(\color{blue}{\left(z + \left(-z \cdot \log t\right)\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   +-commutative [=>] 99.8	\[ x + \left(\color{blue}{\left(\left(-z \cdot \log t\right) + z\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   neg-sub0 [=>] 99.8	\[ x + \left(\left(\color{blue}{\left(0 - z \cdot \log t\right)} + z\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   associate-+l- [=>] 99.8	\[ x + \left(\color{blue}{\left(0 - \left(z \cdot \log t - z\right)\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   associate-+l- [=>] 99.8	\[ x + \color{blue}{\left(0 - \left(\left(z \cdot \log t - z\right) - \left(\left(a - 0.5\right) \cdot b + y\right)\right)\right)} \]
   sub0-neg [=>] 99.8	\[ x + \color{blue}{\left(-\left(\left(z \cdot \log t - z\right) - \left(\left(a - 0.5\right) \cdot b + y\right)\right)\right)} \]

3. Final simplification 99.9%

   \[\leadsto x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]

Alternatives

Alternative 1
Accuracy	88.0%
Cost	9561

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ t_2 := \left(t_1 + \left(x + z\right)\right) - z \cdot \log t\\ t_3 := \left(x + y\right) + t_1\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+170}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -6 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\left(x + y\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+45} \lor \neg \left(t_1 \leq 10^{+150}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Accuracy	88.7%
Cost	7753

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;t_1 \leq -6 \cdot 10^{+46} \lor \neg \left(t_1 \leq 4 \cdot 10^{+55}\right):\\ \;\;\;\;\left(x + y\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + z \cdot \left(1 - \log t\right)\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	7360

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

Alternative 4
Accuracy	86.2%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+166} \lor \neg \left(z \leq 1.95 \cdot 10^{+191}\right):\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 5
Accuracy	86.1%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+168}:\\ \;\;\;\;x + \left(z - z \cdot \log t\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+191}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \end{array} \]

Alternative 6
Accuracy	86.1%
Cost	7112

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+168}:\\ \;\;\;\;y + t_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+191}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + t_1\\ \end{array} \]

Alternative 7
Accuracy	83.0%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+248} \lor \neg \left(z \leq 6 \cdot 10^{+226}\right):\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 8
Accuracy	83.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+248}:\\ \;\;\;\;z - z \cdot \log t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+226}:\\ \;\;\;\;\left(x + y\right) + \left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \end{array} \]

Alternative 9
Accuracy	42.7%
Cost	1361

\[\begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+22}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{elif}\;x + y \leq 10^{+126} \lor \neg \left(x + y \leq 5 \cdot 10^{+200}\right):\\ \;\;\;\;y + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]

Alternative 10
Accuracy	47.2%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;x + y \leq 2 \cdot 10^{-98}:\\ \;\;\;\;x + \left(a + -0.5\right) \cdot b\\ \mathbf{elif}\;x + y \leq 10^{+126} \lor \neg \left(x + y \leq 5 \cdot 10^{+200}\right):\\ \;\;\;\;y + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]

Alternative 11
Accuracy	60.6%
Cost	1097

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+188} \lor \neg \left(t_1 \leq 2 \cdot 10^{+163}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12
Accuracy	29.7%
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-41}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-79}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-120}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq -3.35 \cdot 10^{-202}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-269}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 13
Accuracy	31.0%
Cost	852

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.3 \cdot 10^{-25}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-115}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{-201}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 14
Accuracy	51.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+22}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 15
Accuracy	42.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+22}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + a \cdot b\\ \end{array} \]

Alternative 16
Accuracy	53.1%
Cost	720

\[\begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+170}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;b \leq 240000000000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+188}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \]

Alternative 17
Accuracy	52.4%
Cost	708

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -2 \cdot 10^{-150}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 18
Accuracy	75.8%
Cost	576

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

Alternative 19
Accuracy	31.6%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 20
Accuracy	25.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023140 
(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)))