Beckmann Distribution sample, tan2theta, alphax == alphay

Average Error: 13.3 → 0.3

Time: 19.4s

Precision: binary64

Cost: 6848

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

↓

\[\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) \]

FPCore

(FPCore (alpha u0)
 :precision binary64
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

↓

(FPCore (alpha u0) :precision binary64 (* (- alpha) (* alpha (log1p (- u0)))))

C

double code(double alpha, double u0) {
	return (-alpha * alpha) * log((1.0 - u0));
}

↓

double code(double alpha, double u0) {
	return -alpha * (alpha * log1p(-u0));
}

Java

public static double code(double alpha, double u0) {
	return (-alpha * alpha) * Math.log((1.0 - u0));
}

↓

public static double code(double alpha, double u0) {
	return -alpha * (alpha * Math.log1p(-u0));
}

Python

def code(alpha, u0):
	return (-alpha * alpha) * math.log((1.0 - u0))

↓

def code(alpha, u0):
	return -alpha * (alpha * math.log1p(-u0))

Julia

function code(alpha, u0)
	return Float64(Float64(Float64(-alpha) * alpha) * log(Float64(1.0 - u0)))
end

↓

function code(alpha, u0)
	return Float64(Float64(-alpha) * Float64(alpha * log1p(Float64(-u0))))
end

Wolfram

code[alpha_, u0_] := N[(N[((-alpha) * alpha), $MachinePrecision] * N[Log[N[(1.0 - u0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, u0_] := N[((-alpha) * N[(alpha * N[Log[1 + (-u0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.3

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
   Proof

Alternatives

Alternative 1
Error	13.5
Cost	1216

\[u0 \cdot \left(\alpha \cdot \alpha\right) + \left(\left(u0 \cdot u0\right) \cdot \left(0.3333333333333333 \cdot u0 + 0.5\right)\right) \cdot \left(\alpha \cdot \alpha\right) \]

Alternative 2
Error	13.5
Cost	960

\[\left(u0 - \left(u0 \cdot u0\right) \cdot \left(-0.5 + -0.3333333333333333 \cdot u0\right)\right) \cdot \left(\alpha \cdot \alpha\right) \]

Alternative 3
Error	20.4
Cost	704

\[\left(\alpha \cdot \alpha\right) \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right) \]

Alternative 4
Error	20.4
Cost	704

\[\left(u0 \cdot \left(\alpha - \left(u0 \cdot \alpha\right) \cdot -0.5\right)\right) \cdot \alpha \]

Alternative 5
Error	35.6
Cost	320

\[\left(\alpha \cdot \alpha\right) \cdot u0 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary64
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))