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

Average Accuracy: 99.8% → 99.9%

Time: 21.0s

Precision: binary64

Cost: 13632

Math

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

↓

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

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 (- (+ y (+ x z)) (* 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, ((y + (x + z)) - (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(Float64(y + Float64(x + z)) - 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[(N[(y + N[(x + z), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $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%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a + -0.5, b, \left(y + \left(x + z\right)\right) - z \cdot \log t\right)} \]
   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	\[ \mathsf{fma}\left(a + -0.5, b, \left(\color{blue}{\left(y + x\right)} + z\right) - z \cdot \log t\right) \]
   associate-+l+ [=>] 99.9	\[ \mathsf{fma}\left(a + -0.5, b, \color{blue}{\left(y + \left(x + z\right)\right)} - z \cdot \log t\right) \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	91.7%
Cost	7496

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{+94}:\\ \;\;\;\;\left(y + t_1\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+125}:\\ \;\;\;\;\left(y + x\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) + \left(z - z \cdot \log t\right)\right) + -0.5 \cdot b\\ \end{array} \]

Alternative 2
Accuracy	90.8%
Cost	7364

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;z \leq -6.4 \cdot 10^{+94}:\\ \;\;\;\;\left(y + t_1\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+125}:\\ \;\;\;\;\left(y + x\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;y + \left(\left(x + z\right) - z \cdot \log t\right)\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	7360

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

Alternative 4
Accuracy	90.1%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+85} \lor \neg \left(z \leq 10^{+126}\right):\\ \;\;\;\;y + \left(\left(x + z\right) - z \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + b \cdot \left(a - 0.5\right)\\ \end{array} \]

Alternative 5
Accuracy	89.3%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+94}:\\ \;\;\;\;-0.5 \cdot b + \left(y + z \cdot \left(1 - \log t\right)\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+125}:\\ \;\;\;\;\left(y + x\right) + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(\left(x + z\right) - z \cdot \log t\right)\\ \end{array} \]

Alternative 6
Accuracy	85.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+88} \lor \neg \left(z \leq 9.2 \cdot 10^{+185}\right):\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + b \cdot \left(a - 0.5\right)\\ \end{array} \]

Alternative 7
Accuracy	86.0%
Cost	7112

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+95}:\\ \;\;\;\;y + t_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+185}:\\ \;\;\;\;\left(y + x\right) + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_1\\ \end{array} \]

Alternative 8
Accuracy	84.0%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+128} \lor \neg \left(z \leq 1.1 \cdot 10^{+215}\right):\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + b \cdot \left(a - 0.5\right)\\ \end{array} \]

Alternative 9
Accuracy	48.6%
Cost	1361

\[\begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{-76}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{-34} \lor \neg \left(y + x \leq 10^{+17}\right) \land y + x \leq 10^{+122}:\\ \;\;\;\;b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Accuracy	42.1%
Cost	1361

\[\begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{-76}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;y + x \leq 6 \cdot 10^{-51} \lor \neg \left(y + x \leq 10^{+17}\right) \land y + x \leq 10^{+122}:\\ \;\;\;\;b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]

Alternative 11
Accuracy	57.0%
Cost	1229

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;y + x \leq 6 \cdot 10^{-51}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;y + x \leq 10^{+17} \lor \neg \left(y + x \leq 10^{+122}\right):\\ \;\;\;\;\left(y + x\right) + -0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	47.9%
Cost	1101

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;y + x \leq 6 \cdot 10^{-51}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;y + x \leq 10^{+17} \lor \neg \left(y + x \leq 10^{+122}\right):\\ \;\;\;\;y + -0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Accuracy	60.3%
Cost	1097

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+115} \lor \neg \left(t_1 \leq 10^{+101}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 14
Accuracy	30.5%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-210}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-146}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;y \leq 1000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+101}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 15
Accuracy	52.4%
Cost	720

\[\begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+108}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;b \leq -1.9 \cdot 10^{+17}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-10}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{+178}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \]

Alternative 16
Accuracy	52.6%
Cost	708

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;y + x \leq -1 \cdot 10^{-180}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 17
Accuracy	76.1%
Cost	576

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

Alternative 18
Accuracy	31.8%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 19
Accuracy	23.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(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)))